text: Note that 3 (cid:107) (cid:107) |v | ≤ v $ 1 3 . (12) 4 δ3 3 So v $ (cid:107) (cid:107) ≤ v 2 1 3 . v δ1δ3 $ That is, (cid:107) (cid:107) F− ∗ F− ≤ v 2 ( 1(v)) ( 1(v)) ( 1 )1. δ δ 1 3 Taking the norm on both sides, we have (cid:107) (cid:107) (cid:107) F− ∗ F− (cid:107) ≤ v 2 ( 1(v)) ( 1(v)) ∞ 1 . δ δ 1 3 Hence (cid:107) (cid:107) (cid:107)F− (cid:107) ≤ √v 1(v) ∞ 1 . δ δ 1 3 F F− ∗ ∗ (cid:107) (cid:107) (cid:107) ∗(cid:107) Recall that (v) = ( 1(v )) and v = v , we obtain that 1 1 (cid:107) (cid:107) (cid:107)F (cid:107) ≤ √v (v) ∞ 1 . δ δ 1 3 By symmetry, we have (cid:107) (cid:107) (cid:107)F (cid:107) ≤ √v (v) ∞ 1 . δ δ 2 4 Therefore (cid:107) (cid:107) (cid:112) (cid:112) (cid:107)F (cid:107) ≤ v (v) ∞ 1 , (cid:80) δ (cid:80) 0 { } where δ = max δ δ , δ δ . 0 1 3 2 4 (cid:107) (cid:107) (cid:107) (cid:107) If x is arbitrary in P ±, let x = λ v be the rank-one decomposition, then x = λ v . 2, (cid:107) k(cid:107) k k 1 k k k 1 We have proved that (cid:107)F(v )(cid:107)∞ ≤ (cid:88)vk 1 for the rank-o(cid:88)ne partial isometry v , so k δ k 0 (cid:107) (cid:107) (cid:107) (cid:107) (cid:107)F (cid:107) ≤ (cid:107)F (cid:107) ≤ v x (x) ∞ λ (v ) ∞ λ k 1 = 1 . (13) k k k δ δ 0 0 k k 34 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Note

text: that

label: equation

text: |v

text: v

text: δ3

text: So

text: v

text: v

text: v

text: δ1δ3

text: That

text: is,

text: F−

text: F−

text: v

text: δ

text: δ

text: Taking

text: the

text: norm

text: on

text: both

text: sides,

text: we

text: have

text: F−

text: F−

text: v

text: δ

text: δ

text: Hence

text: √v

text: δ

text: δ

text: F

text: F−

text: Recall

text: that

text: (v)

text: and

text: v

text: v

text: we

text: obtain

text: that

text: √v

text: (v)

text: δ

text: δ

text: By

text: symmetry,

text: we

text: have

text: √v

text: (v)

text: δ

text: δ

text: Therefore

text: v

text: (v)

text: δ

text: where

text: δ

text: max

text: δ

text: δ

text: δ

text: δ

text: If

text: x

text: is

text: arbitrary

text: in

text: P

text: let

text: x

text: λ

text: v

text: be

text: the

text: rank-one

text: decomposition,

text: then

text: x

text: λ

text: v

text: k

text: k

text: k

text: k

text: k

text: We

text: have

text: proved

text: that

text: for

text: the

text: partial

text: isometry

text: v

text: so

text: k

text: δ

text: k

text: v

text: x

text: (x)

text: λ

text: (v

text: λ

text: k

text: k

text: k

text: k

text: δ

text: δ

text: k

text: k

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: Notation 7.4. We call this general version of 2-box x extremal if it satisﬁes the equality (cid:107)F (cid:107) 1 (cid:107) (cid:107) (x) ∞ = x δ 1 0 for this general Fourier transform. √ ∈ ⊗ Lemma 7.5. Suppose x PP4⊗P3 is extremal. If δ = δ δ , then there √are subprojections Q P1 P2 ⊗ 0 1⊗3 i of P with trace δ , for i = 1, 3, such that x = (Q P )x(P Q ); If δ = δ δ , then there are i i 1 2 4 3 ⊗ 0 ⊗2 4 subprojections Q of P with trace δ , for i = 2, 4, such that x = (P Q )x(Q P ). i i i √ 1 2 4 3 Proof. Suppose x is extremal. When δ = δ δ , if x is a rank-one partial isometry, then Equation 0 1 3 (cid:80) (11), (12) hold in the proof of 7.3. So there are subprojections Q of P with trace δ , for i = 1, 3, ⊗ ⊗ i i i such that x = (Q P )x(P Q ). 1 2 4 3 If x is arbitrary, let x = λ v be the rank-one decomposition, then each v is a rank-one k k k k extremal partial isometry. Without loss of generality, we take v and v , then there are subprojections (cid:48) ⊗ 1 ⊗ 2 (cid:48) ⊗ ⊗ (cid:48) Q , Q of P with trace δ , for i = 1, 3, such that v = (Q 1)v (1 Q ); v = (Q 1)v (1 Q ). It i i i i (cid:48) 1 1 1 3 2 1 2 3 is enough to show that Q = Q . Since x is extremal, from Equation (13) in the proof of 7.3, we have i i (cid:107)F F (cid:107) (cid:107)F (cid:107) (cid:107)F (cid:107) (v ) + (v ) ∞ = (v ) ∞ + (v ) ∞. 1 2 1 2 Taking the square on both sides, (cid:107) F F ∗ F F (cid:107) (cid:107)F (cid:107) (cid:107)F (cid:107) ( (v ) + (v )) ( (v ) + (v )) ∞ = ( (v ) ∞ + (v ) ∞)2. 1 2 1 2 1 2 So (cid:107)|F | |F | (cid:107) (cid:107)F (cid:107) (cid:107)F (cid:107) (v ) 2 + (v ) 2 ∞ = (v ) 2∞ + (v ) 2∞. 1 2 1 1 Then (cid:107)(cid:107)F (cid:107) ⊗ (cid:107)F (cid:107) (cid:48) ⊗ (cid:107) ≥ (cid:107)F (cid:107) (cid:107)F (cid:107) (v ) 2∞(Q P ) + (v ) 2∞(Q P ) ∞ (v ) 2∞ + (v ) 2∞. 1 1 4 2 1 4 1 1 So (cid:107)(cid:107)F (cid:107) (cid:107)F (cid:107) (cid:48) (cid:107) ≥ (cid:107)F (cid:107) (cid:107)F (cid:107) (v ) 2∞Q + (v ) 2∞Q ∞ (v ) 2∞ + (v ) 2∞. 1 1 2 1 1 1 (cid:48) (cid:48) That implies Q = Q . Similarl√y Q = Q . 1 1 3 3 The proof is similar if δ = δ δ . 0 2 4 ∈ ⊗ ∈ ⊗ (cid:48) ∗ ∗ For x PPP14⊗PP23, y PPP23⊗PP14(cid:48) , let us deﬁne the coproduct PP32, simply denoted by , as follows $ P 3 ∗ x y x y = $ $ . P 2 $ By a similar argument in Section and the technique in the proof of Theorem 7.3, we have the following Young’s inequality. 35 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Notation

label: paragraph

text: We

text: call

text: this

text: general

text: version

text: of

text: x

text: extremal

text: if

text: it

text: satisﬁes

text: the

text: equality

text: (x)

text: x

text: δ

text: for

text: this

text: general

text: Fourier

text: transform.

text: Lemma

text: Suppose

text: x

text: is

text: extremal.

text: If

text: δ

text: δ

text: δ

text: then

text: there

text: √are

text: subprojections

text: Q

text: i

text: of

text: P

text: with

text: trace

text: δ

text: for

text: i

text: such

text: that

text: x

text: (Q

text: P

text: )x(P

text: Q

text: If

text: δ

text: δ

text: δ

text: then

text: there

text: are

text: i

text: i

text: ⊗2

text: subprojections

text: Q

text: of

text: P

text: with

text: trace

text: δ

text: for

text: i

text: such

text: that

text: x

text: (P

text: Q

text: )x(Q

text: P

text: i

text: i

text: i

text: Proof.

text: Suppose

text: x

text: is

text: extremal.

text: When

text: δ

text: δ

text: δ

text: if

text: x

text: is

text: a

text: rank-one

text: partial

text: isometry,

text: then

text: Equation

text: hold

text: in

text: the

text: proof

text: of

text: So

text: there

text: are

text: subprojections

text: Q

text: of

text: P

text: with

text: trace

text: δ

text: for

text: i

text: i

text: i

text: i

text: such

text: that

text: x

text: (Q

text: P

text: )x(P

text: Q

text: If

text: x

text: is

text: arbitrary,

text: let

text: x

text: λ

text: v

text: be

text: the

text: rank-one

text: decomposition,

text: then

text: each

text: v

text: is

text: a

text: rank-one

text: k

text: k

text: k

text: k

text: extremal

text: partial

text: isometry.

text: Without

text: loss

text: of

text: generality,

text: we

text: take

text: v

text: and

text: v

text: then

text: there

text: are

text: subprojections

text: Q

text: Q

text: of

text: P

text: with

text: trace

text: δ

text: for

text: i

text: such

text: that

text: v

text: (Q

text: Q

text: v

text: (Q

text: Q

text: It

text: i

text: i

text: i

text: i

text: is

text: enough

text: to

text: show

text: that

text: Q

text: Q

text: Since

text: x

text: is

text: extremal,

text: from

text: Equation

text: in

text: the

text: proof

text: of

text: we

text: have

text: i

text: i

text: F

text: (v

text: (v

text: (v

text: (v

text: Taking

text: the

text: square

text: on

text: both

text: sides,

text: F

text: F

text: F

text: F

text: (v

text: (v

text: (v

text: (v

text: (v

text: (v

text: So

text: |F

text: (v

text: (v

text: (v

text: 2∞

text: (v

text: Then

text: (v

text: P

text: (v

text: P

text: (v

text: 2∞

text: (v

text: So

text: (v

text: (v

text: (v

text: 2∞

text: (v

text: That

text: implies

text: Q

text: Q

text: Similarl√y

text: Q

text: Q

text: The

text: proof

text: is

text: similar

text: if

text: δ

text: δ

text: δ

text: For

text: x

text: y

text: let

text: us

text: deﬁne

text: the

text: coproduct

text: simply

text: denoted

text: by

text: as

text: follows

text: P

text: x

text: y

text: x

text: y

text: P

text: By

text: a

text: similar

text: argument

text: in

text: Section

text: and

text: the

text: technique

text: in

text: the

text: proof

text: of

text: Theorem

text: we

text: have

text: the

text: following

text: Young’s

text: inequality.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ For the group case, δ x y is the convolution of functions of the group. By the Hausdorﬀ-Young inequality in subfactor planar algebras, we are able to give the following uncertainty principles. Main Theorem 1. [See Theorem 5.2, 5.5] Suppose x is a nonzero 2-box in an irreducible subfactor planar algebra. Then we have the Hirschman-Beckner uncertainty principle, | | |F | ≥ (cid:107) (cid:107) − (cid:107) (cid:107) H( x 2) + H( (x) 2) x (2 log δ 4 log x ), 2 2 · | | − | | | | where H( ) is the von Neumann entropy, H( x 2) = tr ( x 2 log x 2); and the Donoho-Stark 2 uncertainty principle, S S F ≥ (x) ( (x)) δ2, S where (x) is the trace of the range projection of x. Taking the derivative of the Hausdorﬀ-Young inequality with respect to p at p = 2, we derive − S ≥ | | Hirschman-Beckner uncertainty principle. By the concavity of t log t, we have log (x) log H( x 2), (cid:107) (cid:107) when x = 1. Then we obtain the Donoho-Stark uncertainty principle. 2 We apply the noncommutative version of O¨zaydm and Przebinda [24] to show the minimizers of the two uncertainty principles coincide. To characterize the minimizers of Donoho-Stark uncertainty principle, we apply the noncommutative version of Tao’s proof in [29]. We show that the minimizers are extremal bi-partial isometries (see Deﬁnition 6.1 and Theorem 6.4). Extremal bi-partial isometries are convenient to work with theoretically, but not convenient to construct. Recall that the minimizer of uncertainty principles of ﬁnite abelian groups is given by a nonzero constant multiple of a translation and a modulation of the indicator function of its subgroup. The original technique for ﬁnite abelian groups is not enough to give this kind of characterization for the 2-box space, because the 2-box space may not be a commutative algebra and there is no translation or modulation to shift the minimizer to a good position. Fortunately Bisch and Jones introduced biprojections [2, 4] which generalized the indicator functions of subgroups. We expect to construct minimizers with the knowledge of biprojections. We prove that an extremal bi-partial isometry x F is uniquely determined by the range projections of x and its Fourier transform (x) (see Theorem 6.16). Moreover, two range projections are shifts of a pair of biprojections (see Deﬁnition 6.5), a generalization of the translation or the modulataion of the indicator function of a subgroup. To construct such an extremal bi-partial isometry, we introduce the notion of a bi-shift of a biprojection (see Deﬁnition 6.6), a generalization of a translation and a modulation of the indicator function of a subgroup. Bi-shifts of biprojections are easy to construct. Main Theorem 2. [See Theorem 6.4,6.13 and Proposition 6.10] For a nonzero 2-box x in an irreducible subfactor planar algebra, the following statements are equivalent: S S F (1) (x) ( (x)) = δ2; | | |F | (cid:107) (cid:107) − (cid:107) (cid:107) (2) H( x 2) + H( (x) 2) = x (2 log δ 4 log x ); 2 2 (3) x is an extremal bi-partial isometry; F− (3’) x is a partial isometry and 1(x) is extremal; (4) x is a bi-shift of a biprojection, 6

text: For

text: the

text: group

text: case,

text: δ

text: x

text: y

text: is

text: the

text: convolution

text: of

text: functions

text: of

text: the

text: group.

text: By

text: the

text: Hausdorﬀ-Young

text: inequality

text: in

text: subfactor

text: planar

text: algebras,

text: we

text: are

text: able

text: to

text: give

text: the

text: following

text: uncertainty

text: principles.

text: Main

text: Theorem

text: [See

text: Theorem

text: Suppose

text: x

text: is

text: a

text: nonzero

text: in

text: an

text: irreducible

text: subfactor

text: planar

text: algebra.

text: Then

text: we

text: have

text: the

text: Hirschman-Beckner

text: uncertainty

text: principle,

text: |F

text: H(

text: x

text: H(

text: (x)

text: x

text: log

text: δ

text: log

text: x

text: where

text: H(

text: is

text: the

text: von

text: Neumann

text: entropy,

text: H(

text: x

text: tr

text: x

text: log

text: x

text: and

text: the

text: Donoho-Stark

text: uncertainty

text: principle,

text: S

text: S

text: F

text: (x)

text: (x))

text: S

text: where

text: (x)

text: is

text: the

text: trace

text: of

text: the

text: range

text: projection

text: of

text: x.

text: Taking

text: the

text: derivative

text: of

text: the

text: Hausdorﬀ-Young

text: inequality

text: with

text: respect

text: to

text: p

text: at

text: p

text: we

text: derive

text: S

text: Hirschman-Beckner

text: uncertainty

text: principle.

text: By

text: the

text: concavity

text: of

text: t

text: log

text: t,

text: we

text: have

text: log

text: (x)

text: log

text: H(

text: x

text: when

text: x

text: Then

text: we

text: obtain

text: the

text: Donoho-Stark

text: uncertainty

text: principle.

text: We

text: apply

text: the

text: noncommutative

text: version

text: of

text: O¨zaydm

text: and

text: Przebinda

text: to

text: show

text: the

text: minimizers

text: of

text: the

text: two

text: uncertainty

text: principles

text: coincide.

text: To

text: characterize

text: the

text: minimizers

text: of

text: Donoho-Stark

text: uncertainty

text: principle,

text: we

text: apply

text: the

text: noncommutative

text: version

text: of

text: Tao’s

text: proof

text: in

text: We

text: show

text: that

text: the

text: minimizers

text: are

text: extremal

text: bi-partial

text: isometries

text: (see

text: Deﬁnition

text: and

text: Theorem

text: Extremal

text: bi-partial

text: isometries

text: are

text: convenient

text: to

text: work

text: with

text: theoretically,

text: but

text: not

text: convenient

text: to

text: construct.

text: Recall

text: that

text: the

text: minimizer

text: of

text: uncertainty

text: principles

text: of

text: ﬁnite

text: abelian

text: groups

text: is

text: given

text: by

text: a

text: nonzero

text: constant

text: multiple

text: of

text: a

text: translation

text: and

text: a

text: modulation

text: of

text: the

text: indicator

text: function

text: of

text: its

text: subgroup.

text: The

text: original

text: technique

text: for

text: ﬁnite

text: abelian

text: groups

text: is

text: not

text: enough

text: to

text: give

text: this

text: kind

text: of

text: characterization

text: for

text: the

text: space,

text: because

text: the

text: space

text: may

text: not

text: be

text: a

text: commutative

text: algebra

text: and

text: there

text: is

text: no

text: translation

text: or

text: modulation

text: to

text: shift

text: the

text: minimizer

text: to

text: a

text: good

text: position.

text: Fortunately

text: Bisch

text: and

text: Jones

text: introduced

text: biprojections

text: which

text: generalized

text: the

text: indicator

text: functions

text: of

text: subgroups.

text: We

text: expect

text: to

text: construct

text: minimizers

text: with

text: the

text: knowledge

text: of

text: biprojections.

text: We

text: prove

text: that

text: an

text: extremal

text: bi-partial

text: isometry

text: x

text: F

text: is

text: uniquely

text: determined

text: by

text: the

text: range

text: projections

text: of

text: x

text: and

text: its

text: Fourier

text: transform

text: (x)

text: (see

text: Theorem

text: Moreover,

text: two

text: range

text: projections

text: are

text: shifts

text: of

text: a

text: pair

text: of

text: biprojections

text: (see

text: Deﬁnition

text: a

text: generalization

text: of

text: the

text: translation

text: or

text: the

text: modulataion

text: of

text: the

text: indicator

text: function

text: of

text: a

text: subgroup.

text: To

text: construct

text: such

text: an

text: extremal

text: bi-partial

text: isometry,

text: we

text: introduce

text: the

text: notion

text: of

text: a

text: bi-shift

text: of

text: a

text: biprojection

text: (see

text: Deﬁnition

text: a

text: generalization

text: of

text: a

text: translation

text: and

text: a

text: modulation

text: of

text: the

text: indicator

text: function

text: of

text: a

text: subgroup.

text: Bi-shifts

text: of

text: biprojections

text: are

text: easy

text: to

text: construct.

text: Main

text: Theorem

text: [See

text: Theorem

text: and

text: Proposition

text: For

text: a

text: nonzero

text: x

text: in

text: an

text: irreducible

text: subfactor

text: planar

text: algebra,

text: the

text: following

text: statements

text: are

text: equivalent:

text: S

text: S

text: F

text: (x)

text: (x))

text: |F

text: H(

text: x

text: H(

text: (x)

text: x

text: log

text: δ

text: log

text: x

text: x

text: is

text: an

text: extremal

text: bi-partial

text: isometry;

text: F−

text: x

text: is

text: a

text: partial

text: isometry

text: and

text: is

text: extremal;

text: x

text: is

text: a

text: bi-shift

text: of

text: a

text: biprojection,


text: Theorem 2.11 says that in particular for semi-toric systems, under this mild assumption that is topological stability on the leaves, a critical leaf Λ is diﬀeomorphic to a product of the “simplest” regular, elliptic and focus-focus leaves one can ﬁnd. Remark 2.12. Assertion 3. of Theorem 2.11 explains that we do not have a symplectomorphism in the assertion 2. Also, one should notice that in Theorem 2.11, it is only because we made the Assumption 2.10 that the singularity is leaf-wise diﬀeomorphic to an almost-direct product of simple sigularities. For instance, were there more than one pinch on the singuarity, one could only guarantee the existence of an homeomorphism between the two. This theorem is the last generalisation of Eliasson’s point-wise normal form. As of now, we shall speak of Eliasson-Miranda-Zung normal form and precise if necessary what type of normal form we are working on. 2.4. Stratiﬁcation by Williamson index The stratiﬁcation of M by the rank of the moment map is an important feature in the study of toric system in the theorem of Atiyah - Guillemin & Steinberg. The following results are consequences of Theorem 2.11, and of its proof in [Zun96]. By stratiﬁcation, we mean here the following S I Deﬁnition 2.13. A stratiﬁed manifold is a triplet (M, , ) with Mn topo- S logical manifold is a ﬁnite partition = (S ) ∈I of Mn indexed by an ordered I (cid:22) i i set ( , ) such that • ∈ I ∈ I Decomposition: for i the S ’s are smooth manifolds and for i, j , (cid:22) ⊆ i i j if and only if S S . i j • Splitting condition: by induction over the dimension of the stratiﬁed manifold n. ∈ R ⊂ If x S , for a neighborhood U of x in n, there exists a disk D R i C x i dim(Si) and a cone (J) = over a (n dim S 1)-dimensional stratiﬁed − − ≤ i − smooth manifold J (and (n dim S 1) n 1 ) such that: U and × C i x D (J) are isomorphic as stratiﬁed manifolds. i The deﬁnition above is consistent, since straﬁcations of manifolds of dimension 0 and 1 are obvious. Usually, one deﬁnes stratiﬁcations over larger classes of objects that are not regular enough for the given topology, while the strata are. The example to remember is the cone, or the manifolds with corners: they won’t be smooth manifold but their strata will be. 11 ##LTLine##

text: Theorem

font: CMRoman-Regular

text: says

text: that

text: in

text: particular

text: for

text: semi-toric

text: systems,

text: under

text: this

text: mild

text: assumption

text: that

text: is

text: topological

text: stability

text: on

text: the

text: leaves,

text: a

text: critical

text: leaf

text: Λ

text: is

text: diﬀeomorphic

text: to

text: a

text: product

text: of

text: the

text: “simplest”

text: regular,

text: elliptic

text: and

text: focus-focus

text: leaves

text: one

text: can

text: ﬁnd.

text: Remark

font: CMRoman-Bold

text: Assertion

font: CMRoman-Italic

text: of

text: Theorem

text: explains

text: that

text: we

text: do

text: not

text: have

text: a

text: symplectomorphism

text: in

text: the

text: assertion

text: Also,

text: one

text: should

text: notice

text: that

text: in

text: Theorem

text: it

text: is

text: only

text: because

text: we

text: made

text: the

text: Assumption

text: that

text: the

text: singularity

text: is

text: leaf-wise

text: diﬀeomorphic

text: to

text: an

text: almost-direct

text: product

text: of

text: simple

text: sigularities.

text: For

text: instance,

text: were

text: there

text: more

text: than

text: one

text: pinch

text: on

text: the

text: singuarity,

text: one

text: could

text: only

text: guarantee

text: the

text: existence

text: of

text: an

text: homeomorphism

text: between

text: the

text: two.

text: This

text: theorem

text: is

text: the

text: last

text: generalisation

text: of

text: Eliasson’s

text: point-wise

text: normal

text: form.

text: As

text: of

text: now,

text: we

text: shall

text: speak

text: of

text: Eliasson-Miranda-Zung

text: normal

text: form

text: and

text: precise

text: if

text: necessary

text: what

text: type

text: of

text: normal

text: form

text: we

text: are

text: working

text: on.

text: Stratiﬁcation

text: by

text: Williamson

text: index

text: The

text: stratiﬁcation

text: of

text: M

text: by

text: the

text: rank

text: of

text: the

text: moment

text: map

text: is

text: an

text: important

text: feature

text: in

text: the

text: study

text: of

text: toric

text: system

text: in

text: the

text: theorem

text: of

text: Atiyah

text: Guillemin

text: Steinberg.

text: The

text: following

text: results

text: are

text: consequences

text: of

text: Theorem

text: and

text: of

text: its

text: proof

text: in

text: By

text: stratiﬁcation,

text: we

text: mean

text: here

text: the

text: following

text: S

text: I

text: Deﬁnition

text: A

text: stratiﬁed

text: manifold

text: is

text: a

text: triplet

text: (M,

text: with

text: Mn

text: topo-

text: S

text: logical

text: manifold

text: is

text: a

text: ﬁnite

text: partition

text: (S

text: ∈I

text: of

text: Mn

text: indexed

text: by

text: an

text: ordered

text: I

text: i

text: i

text: set

text: such

text: that

text: I

text: I

text: Decomposition:

text: for

text: i

text: the

text: S

text: ’s

text: are

text: smooth

text: manifolds

text: and

text: for

text: i,

text: j

text: i

text: i

text: j

text: if

text: and

text: only

text: if

text: S

text: S

text: i

text: j

text: Splitting

text: condition:

text: by

text: induction

text: over

text: the

text: dimension

text: of

text: the

text: stratiﬁed

text: manifold

text: n.

text: R

text: If

text: x

text: S

text: for

text: a

text: neighborhood

text: U

text: of

text: x

text: in

text: n,

text: there

text: exists

text: a

text: disk

text: D

text: R

text: i

text: C

text: x

text: i

text: dim(Si)

text: and

text: a

text: cone

text: (J)

text: over

text: a

text: (n

text: dim

text: S

text: stratiﬁed

text: i

text: smooth

text: manifold

text: J

text: (and

text: (n

text: dim

text: S

text: n

text: such

text: that:

text: U

text: and

text: C

text: i

text: x

text: D

text: (J)

text: are

text: isomorphic

text: as

text: stratiﬁed

text: manifolds.

text: i

text: The

text: deﬁnition

text: above

text: is

text: consistent,

text: since

text: straﬁcations

text: of

text: manifolds

text: of

text: dimension

text: and

text: are

text: obvious.

text: Usually,

text: one

text: deﬁnes

text: stratiﬁcations

text: over

text: larger

text: classes

text: of

text: objects

text: that

text: are

text: not

text: regular

text: enough

text: for

text: the

text: given

text: topology,

text: while

text: the

text: strata

text: are.

text: The

text: example

text: to

text: remember

text: is

text: the

text: cone,

text: or

text: the

text: manifolds

text: with

text: corners:

text: they

text: won’t

text: be

text: smooth

text: manifold

text: but

text: their

text: strata

text: will

text: be.

text: ##LTLine##


text: a point, and we need to show there exists a 3-dimensional stradispace L such that a neighborhood of the focus-focus point is homeomorphic to this point times the cone over L. We can just take the 3-sphere S3 and take the cone over it: it is homeomorphic to the 4-ball, and hence is a neighborhood of a focus-focus point. There is a natural way to reﬁne the stratiﬁcation, by distinguishing the k connected components of the Pk(M). We can note Pkl(M), with l = 1..L( ) an integer labelling the l-th connected components of Pk(M). The new ordering W k set is thus obtained from (F ) by “splitting” each : we consider the couples k k (cid:52) k(cid:48) (cid:48) k (cid:52) k(cid:48) ⊆ ( , l), and write ( , l) ( , l ) if and only if and Pkl Pkl(M). The proof of the stratiﬁcation is exactly the same since the spliting condition is a local condition. Now, if kf = 1, we can take the skeleton of a strata, Pkl(M(cid:54) ). −Here, the skeleton is obtained by patching smoothly pieces of dimension k 2, so we x can extend ωk and Fk such that (P(cid:52)l k, ω(cid:52)k, F(cid:52)k) is an integrable Hamiltonian system on a closed manifold. This system is toric: the singularities are non-degenerate and without hyperbolic component (such component would have to come from the total system) or focus-focus component (because (cid:62) k = 1, such point would be a critical point with k 2 on (M, ω, F ), which is f f impossible). Lastly, since the Williamson type is invariant by the Hamiltonian T ﬂow, Fk yields a k-action on P(cid:52)k(M). x 3. Local models for semi-toric values Eliasson normal form theorem 2.2 and Miranda-Zung theorem 2.4 give us a general understanding of the commutant of a semi-toric system: we have a local model of the image of the moment map near a critical value, that can be precised in the semi-toric case. To do so, we remind ﬁrst these two lemmas. F F(cid:48) Two Lagrangian foliations and are said to be equivalent if there ∈ F (cid:48) ∈ F(cid:48) exists a symplectomorphism such that for each L , ϕ(L) = L . F Lemma 3.1. Let be a singular Liouville foliation given by a momentum → R F(cid:48) map F : M k. Let be a singular Liouville foliation given by a (cid:48) → R momentum map F : N k. If the level sets are locally connected, then for U ⊂ → V ⊂ U every smooth symplectomorphism ϕ : M N where is an open ∈ V (cid:48) ∈ neighborhood of p M, a neighborhood of p = ϕ(p) N, and such that ∗F F(cid:48) ϕ = , R → R (cid:48) (cid:48) there exists a unique local diﬀeomorphism G : ( k, F (p)) ( k, F (p )) such that 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: a

text: point,

text: and

text: we

text: need

text: to

text: show

text: there

text: exists

text: a

text: stradispace

text: L

text: such

text: that

text: a

text: neighborhood

text: of

text: the

text: focus-focus

text: point

text: is

text: homeomorphic

text: to

text: this

text: point

text: times

text: the

text: cone

text: over

text: L.

text: We

text: can

text: just

text: take

text: the

text: and

text: take

text: the

text: cone

text: over

text: it:

text: it

text: is

text: homeomorphic

text: to

text: the

text: and

text: hence

text: is

text: a

text: neighborhood

text: of

text: a

text: focus-focus

text: point.

text: There

text: is

text: a

text: natural

text: way

text: to

text: reﬁne

text: the

text: stratiﬁcation,

text: by

text: distinguishing

text: the

text: k

text: connected

text: components

text: of

text: the

text: Pk(M).

text: We

text: can

text: note

text: Pkl(M),

text: with

text: l

text: an

text: integer

text: labelling

text: the

text: l-th

text: connected

text: components

text: of

text: Pk(M).

text: The

text: new

text: ordering

text: W

text: k

text: set

text: is

text: thus

text: obtained

text: from

text: (F

text: by

text: “splitting”

text: each

text: we

text: consider

text: the

text: couples

text: k

text: k

text: k

text: l),

text: and

text: write

text: l)

text: l

text: if

text: and

text: only

text: if

text: and

text: Pkl

text: Pkl(M).

text: The

text: proof

text: of

text: the

text: stratiﬁcation

text: is

text: exactly

text: the

text: same

text: since

text: the

text: spliting

text: condition

text: is

text: a

text: local

text: condition.

text: Now,

text: if

text: kf

text: we

text: can

text: take

text: the

text: skeleton

text: of

text: a

text: strata,

text: −Here,

text: the

text: skeleton

text: is

text: obtained

text: by

text: patching

text: smoothly

text: pieces

text: of

text: dimension

text: k

text: so

text: we

text: x

text: can

text: extend

text: ωk

text: and

text: Fk

text: such

text: that

text: k,

text: is

text: an

text: integrable

text: Hamiltonian

text: system

text: on

text: a

text: closed

text: manifold.

text: This

text: system

text: is

text: toric:

text: the

text: singularities

text: are

text: non-degenerate

text: and

text: without

text: hyperbolic

text: component

text: (such

text: component

text: would

text: have

text: to

text: come

text: from

text: the

text: total

text: system)

text: or

text: focus-focus

text: component

text: (because

text: k

text: such

text: point

text: would

text: be

text: a

text: critical

text: point

text: with

text: k

text: on

text: (M,

text: ω,

text: F

text: which

text: is

text: f

text: f

text: impossible).

text: Lastly,

text: since

text: the

text: Williamson

text: type

text: is

text: invariant

text: by

text: the

text: Hamiltonian

text: T

text: ﬂow,

text: Fk

text: yields

text: a

text: k-action

text: on

text: x

text: Local

text: models

text: for

text: semi-toric

text: values

text: Eliasson

text: normal

text: form

text: theorem

text: and

text: Miranda-Zung

text: theorem

text: give

text: us

text: a

text: general

text: understanding

text: of

text: the

text: commutant

text: of

text: a

text: semi-toric

text: system:

text: we

text: have

text: a

text: local

text: model

text: of

text: the

text: image

text: of

text: the

text: moment

text: map

text: near

text: a

text: critical

text: value,

text: that

text: can

text: be

text: precised

text: in

text: the

text: semi-toric

text: case.

text: To

text: do

text: so,

text: we

text: remind

text: ﬁrst

text: these

text: two

text: lemmas.

text: F

text: Two

text: Lagrangian

text: foliations

text: and

text: are

text: said

text: to

text: be

text: equivalent

text: if

text: there

text: F

text: exists

text: a

text: symplectomorphism

text: such

text: that

text: for

text: each

text: L

text: ϕ(L)

text: L

text: F

text: Lemma

text: Let

text: be

text: a

text: singular

text: Liouville

text: foliation

text: given

text: by

text: a

text: momentum

text: R

text: map

text: F

text: M

text: k.

text: Let

text: be

text: a

text: singular

text: Liouville

text: foliation

text: given

text: by

text: a

text: R

text: momentum

text: map

text: F

text: N

text: k.

text: If

text: the

text: level

text: sets

text: are

text: locally

text: connected,

text: then

text: for

text: U

text: V

text: U

text: every

text: smooth

text: symplectomorphism

text: ϕ

text: M

text: N

text: where

text: is

text: an

text: open

text: V

text: neighborhood

text: of

text: p

text: M,

text: a

text: neighborhood

text: of

text: p

text: ϕ(p)

text: N,

text: and

text: such

text: that

text: ∗F

text: ϕ

text: R

text: R

text: there

text: exists

text: a

text: unique

text: local

text: diﬀeomorphism

text: G

text: k,

text: F

text: (p))

text: k,

text: F

text: (p

text: such

text: that

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof. of Theorem 3.7 First let’s prove equation 8. ∗ ∗ · − · ζ (q + iq ) = ζ (z¯ z ) = eiθ Xfz¯ e iθ Xfz = z¯ z = q + iq B 1 2 B 1 2 1 2 1 2 1 2 so q , q are preserved. 1 2 ∗ | − · · | | − · · | | | | | ζ Qek = ( e iθ (X1e,.)t ze 2, . . . , e iθ (Xkee,.)t ze 2) = ( ze 2, . . . , ze 2) = Qek B 1 ke 1 ke ∗ · · ζ I = q Xf + Qek (Xe)t + I. B 2 ∗ ∗ ∗ For η, we have: η Qk = (E1E2) Qfk + (Xx) Qxk , so we can treat each action separately. We can also treat E and E separately, as the two 1 2 matrices commutes, and treat only the case when (cid:15)f (respectively (cid:15)f ) is equal − 1 2 to 1, for when (cid:15)f = +1, E = id. i i • − (cid:15)f = 1:         1                  0 1 0 0 x   ξ  xˆ  − 1 −1 1 1 0 0 0 · ξ x ξˆ E = so E 1 = 1 = 1 1 0 0 0 1 1 x ξ xˆ − 2 −2 2 0 0 1 0 ξ x ξˆ 2 2 2 ∗ − − − ˆ ˆ f E q = xˆ ξ + xˆ ξ = ξ x ξ x = q = (cid:15) q , 1 1 1 1 2 2 1 1 2 2 1 1 1 ∗ − − − − ˆ ˆ E q = xˆ ξ xˆ ξ = ξ ( x ) ξ ( x ) = q . 1 2 1 2 2 1 1 2 2 1 2 • −   (cid:15)f = 1: 2     0 0 1 0 0 0 0 1 E = 2 1 0 0 0 0 1 0 0 ∗ ˜ ˜ E q = x˜ ξ + x˜ ξ = x ξ + x ξ = q 2 1 1 1 2 2 2 2 1 1 1 ∗ − − − ˜ ˜ f E q = x˜ ξ x˜ ξ = x ξ x ξ = q = (cid:15) q . 2 2 1 2 2 1 2 1 1 2 2 2 2 23

text: Proof.

text: of

text: Theorem

text: First

text: let’s

text: prove

text: equation

text: ζ

text: (q

text: iq

text: ζ

text: (z¯

text: z

text: eiθ

text: Xfz¯

text: e

text: iθ

text: Xfz

text: z¯

text: z

text: q

text: iq

text: B

text: B

text: so

text: q

text: q

text: are

text: preserved.

text: ζ

text: Qek

text: e

text: iθ

text: ze

text: e

text: iθ

text: (Xkee,.)t

text: ze

text: ze

text: ze

text: Qek

text: B

text: ke

text: ke

text: ζ

text: I

text: q

text: Xf

text: Qek

text: (Xe)t

text: I.

text: B

text: For

text: η,

text: we

text: have:

text: η

text: Qk

text: Qfk

text: (Xx)

text: Qxk

text: so

text: we

text: can

text: treat

text: each

text: action

text: separately.

text: We

text: can

text: also

text: treat

text: E

text: and

text: E

text: separately,

text: as

text: the

text: two

text: matrices

text: commutes,

text: and

text: treat

text: only

text: the

text: case

text: when

text: (respectively

text: is

text: equal

text: to

text: for

text: when

text: E

text: id.

text: i

text: i

text: 0

text: x

text: ξ

text: xˆ

text: −1

text: ξ

text: x

text: ξˆ

text: E

text: so

text: E

text: x

text: ξ

text: xˆ

text: −2

text: ξ

text: x

text: ξˆ

text: ˆ

text: ˆ

text: f

text: E

text: q

text: xˆ

text: ξ

text: xˆ

text: ξ

text: ξ

text: x

text: ξ

text: x

text: q

text: q

text: ˆ

text: ˆ

text: E

text: q

text: xˆ

text: ξ

text: xˆ

text: ξ

text: ξ

text: x

text: ξ

text: x

text: q

text: 0

text: 0

text: E

text: E

text: q

text: x˜

text: ξ

text: x˜

text: ξ

text: x

text: ξ

text: x

text: ξ

text: q

text: f

text: E

text: q

text: x˜

text: ξ

text: x˜

text: ξ

text: x

text: ξ

text: x

text: ξ

text: q

text: q


text: This theorem means that the only symplectic invariants of the local model of a semi-toric critical value are the orientations of the half-spaces given by its elliptic components and its jet in q , q . Conservation of plans orientations 1 2 is only natural: symplectic structure is exactly the algebraic area on speciﬁc plans. Conservation of the jet of the focus-focus value, is a speciﬁcity of semi-toric systems. It is related to the Taylor expansion of action coordinates near a semi-toric critical value (see [Wac15]). 4. Image of moment map for a semi-toric integrable system Now that we have gathered enough results concerning local models, we can prove the principal result of this paper by exhibitng a local-to-global principle. Proof. of Theorem 1.6 Local proof of 1. , 2. and 3.: k Let p be a critical point of Williamson type with k = 1 of a semi-toric f integrable system (M, ω, F ). Applying Theorem 3.3, with the correct system U of local coordinates ϕ in a neighborhood of p, we have a smooth function ∈ Z ◦ ◦ G and a matrix A GL − ( ) such that F ϕ = (G (Qk), A Qˇk). So, the U n 1 1 surface ΓU := Vk( ) is parametrized as follows: let t be here the values of I. (cid:124) (cid:123)(cid:122) (cid:125) ∈ ⊆ R C∞ → R With Theorem 2.14, we know that t D˜ kx. We deﬁne now h (D˜ ) as h˜(t) := G (0, 0, 0, . . . , 0, t). With notations of Theorem 3.3, we set 1     ke  (cid:124) (cid:123)(cid:122) (cid:125)  U ◦ ◦ ◦ | ∈ ΓU := Vk( ) = H˜ (t) = (h˜(t), F x t, Ex t, Xx t) t D . =:S U On Pk( ), F is of rank kx by deﬁnition, and in Theorem 3.3, ﬁrst column (cid:124) (cid:123)(cid:122) (cid:125) R of Jac(G) is (∂ G , 0, .., 0)T , so there exists a linear map S of n such that ◦ ◦q1 1 S ˜t(t) = (S h˜(t), 0, .., 0, t)). 1 ke+1 P This proves points 1.,2. and 3. locally: (ΓU) is the aﬃne space generated ◦ U by Im(h) and Im(T ), and with h := S h˜ on , we have that ΓU is the 1 (cid:54) graph of h on D. Note that since ∂ G (0, .., 0, t) = 0, Im(h) is not in Im(T ). q1 1 Global proof of 1. , 2. and 3.: U ∈ If we call the open set given for a point p M by Theorem 2.11, the {U } p (cid:83) (cid:83) family ∈ is an open covering of M, so we can extract a ﬁnite one of it. If we npowp ﬁMx k, each open s(cid:8)et U(cid:9)k gives a surface Γ , in Vk(M) there is at most a ﬁnite number mk of surfaceis: Vk(M) = mk Vi k(Uk) = mk Γ . We want to show that the covering Uk of Pk(Mi=)1can bei optimii=ze1d iin the i i=1..mk 25

text: This

text: theorem

text: means

text: that

text: the

text: only

text: symplectic

text: invariants

text: of

text: the

text: local

text: model

text: of

text: a

text: semi-toric

text: critical

text: value

text: are

text: the

text: orientations

text: of

text: the

text: half-spaces

text: given

text: by

text: its

text: elliptic

text: components

text: and

text: its

text: jet

text: in

text: q

text: q

text: Conservation

text: of

text: plans

text: orientations

text: is

text: only

text: natural:

text: symplectic

text: structure

text: is

text: exactly

text: the

text: algebraic

text: area

text: on

text: speciﬁc

text: plans.

text: Conservation

text: of

text: the

text: jet

text: of

text: the

text: focus-focus

text: value,

text: is

text: a

text: speciﬁcity

text: of

text: semi-toric

text: systems.

text: It

text: is

text: related

text: to

text: the

text: Taylor

text: expansion

text: of

text: action

text: coordinates

text: near

text: a

text: semi-toric

text: critical

text: value

text: (see

text: Image

text: of

text: moment

text: map

text: for

text: a

text: semi-toric

text: integrable

text: system

text: Now

text: that

text: we

text: have

text: gathered

text: enough

text: results

text: concerning

text: local

text: models,

text: we

text: can

text: prove

text: the

text: principal

text: result

text: of

text: this

text: paper

text: by

text: exhibitng

text: a

text: local-to-global

text: principle.

text: Proof.

text: of

text: Theorem

text: Local

text: proof

text: of

text: and

text: k

text: Let

text: p

text: be

text: a

text: critical

text: point

text: of

text: Williamson

text: type

text: with

text: k

text: of

text: a

text: semi-toric

text: f

text: integrable

text: system

text: (M,

text: ω,

text: F

text: Applying

text: Theorem

text: with

text: the

text: correct

text: system

text: U

text: of

text: local

text: coordinates

text: ϕ

text: in

text: a

text: neighborhood

text: of

text: p,

text: we

text: have

text: a

text: smooth

text: function

text: Z

text: G

text: and

text: a

text: matrix

text: A

text: GL

text: such

text: that

text: F

text: ϕ

text: (G

text: (Qk),

text: A

text: Qˇk).

text: So,

text: the

text: U

text: n

text: surface

text: ΓU

text: Vk(

text: is

text: parametrized

text: as

text: follows:

text: let

text: t

text: be

text: here

text: the

text: values

text: of

text: I.

text: R

text: C∞

text: R

text: With

text: Theorem

text: we

text: know

text: that

text: t

text: D˜

text: kx.

text: We

text: deﬁne

text: now

text: h

text: (D˜

text: as

text: h˜(t)

text: G

text: t).

text: With

text: notations

text: of

text: Theorem

text: we

text: set

text: ke

text: U

text: ΓU

text: Vk(

text: H˜

text: (t)

text: (h˜(t),

text: F

text: x

text: t,

text: Ex

text: t,

text: Xx

text: t)

text: t

text: D

text: =:S

text: U

text: On

text: Pk(

text: F

text: is

text: of

text: rank

text: kx

text: by

text: deﬁnition,

text: and

text: in

text: Theorem

text: ﬁrst

text: column

text: R

text: of

text: Jac(G)

text: is

text: G

text: so

text: there

text: exists

text: a

text: linear

text: map

text: S

text: of

text: n

text: such

text: that

text: S

text: ˜t(t)

text: (S

text: h˜(t),

text: t)).

text: P

text: This

text: proves

text: points

text: and

text: locally:

text: (ΓU)

text: is

text: the

text: aﬃne

text: space

text: generated

text: U

text: by

text: Im(h)

text: and

text: Im(T

text: and

text: with

text: h

text: S

text: h˜

text: on

text: we

text: have

text: that

text: ΓU

text: is

text: the

text: graph

text: of

text: h

text: on

text: D.

text: Note

text: that

text: since

text: G

text: t)

text: Im(h)

text: is

text: not

text: in

text: Im(T

text: Global

text: proof

text: of

text: and

text: U

text: If

text: we

text: call

text: the

text: open

text: set

text: given

text: for

text: a

text: point

text: p

text: M

text: by

text: Theorem

text: the

text: {U

text: p

text: family

text: is

text: an

text: open

text: covering

text: of

text: M,

text: so

text: we

text: can

text: extract

text: a

text: ﬁnite

text: one

text: of

text: it.

text: If

text: we

text: npowp

text: ﬁMx

text: k,

text: each

text: open

text: gives

text: a

text: surface

text: Γ

text: in

text: Vk(M)

text: there

text: is

text: at

text: most

text: a

text: ﬁnite

text: number

text: mk

text: of

text: surfaceis:

text: Vk(M)

text: mk

text: Vi

text: k(Uk)

text: mk

text: Γ

text: We

text: want

text: to

text: show

text: that

text: the

text: covering

text: Uk

text: of

text: bei

text: iin

text: the

text: i


text: [VN00] S. Vu˜ Ngo.c. Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type. Comm. Pure Appl. Math., 53(2):143–217, 2000. [VN03] S. Vu˜ Ngo.c. On semi-global invariants for focus-focus singularities. Topology, 42(2):365–380, 2003. [VN07] S. Vu˜ Ngo.c. Moment polytopes for symplectic manifolds with monodromy. Advances in Mathematics, 208(2):909 – 934, 2007. [VNW13] S. Vu˜ Ngo.c and C. Wacheux. Smooth normal forms for inte- grable hamiltonian systems near a focus-focus singularity. Acta Mathematica Vietnamica, 38(1):107–122, 2013. [Wac15] C. Wacheux. Asymptotics of action variables near semi- toric singularities. Journal of Geometry and Physics, 2015. http://dx.doi.org/10.1016/j.geomphys.2015.07.023. [Wil36] J. Williamson. On the algebraic problem concerning the normal form of linear dynamical systems. American Journal of Mathe- matics, 58(1):141–163, 1936. [Zun96] Nguyen Tien Zung. Symplectic topology of integrable hamiltonian systems. i : Arnold-liouville with singularities. Compos. Math., 101(2):179–215, 1996. [Zun97] Nguyen Tien Zung. A note on focus-focus singularities. Diﬀeren- tial Geometry and its Applications, 7(2):123 – 130, 1997. [Zun02] Nguyen Tien Zung. Another note on focus-focus singularities. Letter in Mathematical Physics, 60(1):87–99, 2002. [Zun03] Nguyen Tien Zung. Symplectic topology of integrable hamiltonian systems, ii: Topological classiﬁcation. Compositio Mathematica, 138(02):125–156, 2003. 29

label: reference

text: S.

text: Vu˜

text: Ngo.c.

text: Bohr-Sommerfeld

text: conditions

text: for

text: integrable

text: systems

text: with

text: critical

text: manifolds

text: of

text: focus-focus

text: type.

text: Comm.

text: Pure

text: Appl.

text: Math.,

text: S.

text: Vu˜

text: Ngo.c.

text: On

text: semi-global

text: invariants

text: for

text: focus-focus

text: singularities.

text: Topology,

text: S.

text: Vu˜

text: Ngo.c.

text: Moment

text: polytopes

text: for

text: symplectic

text: manifolds

text: with

text: monodromy.

text: Advances

text: in

text: Mathematics,

text: S.

text: Vu˜

text: Ngo.c

text: and

text: C.

text: Wacheux.

text: Smooth

text: normal

text: forms

text: for

text: inte-

text: grable

text: hamiltonian

text: systems

text: near

text: a

text: focus-focus

text: singularity.

text: Acta

text: Mathematica

text: Vietnamica,

text: C.

text: Wacheux.

text: Asymptotics

text: of

text: action

text: variables

text: near

text: semi-

text: toric

text: singularities.

text: Journal

text: of

text: Geometry

text: and

text: Physics,

text: J.

text: Williamson.

text: On

text: the

text: algebraic

text: problem

text: concerning

text: the

text: normal

text: form

text: of

text: linear

text: dynamical

text: systems.

text: American

text: Journal

text: of

text: Mathe-

text: matics,

text: Nguyen

text: Tien

text: Zung.

text: Symplectic

text: topology

text: of

text: integrable

text: hamiltonian

text: systems.

text: i

text: Arnold-liouville

text: with

text: singularities.

text: Compos.

text: Math.,

text: Nguyen

text: Tien

text: Zung.

text: A

text: note

text: on

text: focus-focus

text: singularities.

text: Diﬀeren-

text: tial

text: Geometry

text: and

text: its

text: Applications,

text: Nguyen

text: Tien

text: Zung.

text: Another

text: note

text: on

text: focus-focus

text: singularities.

text: Letter

text: in

text: Mathematical

text: Physics,

text: Nguyen

text: Tien

text: Zung.

text: Symplectic

text: topology

text: of

text: integrable

text: hamiltonian

text: systems,

text: ii:

text: Topological

text: classiﬁcation.

text: Compositio

text: Mathematica,


text: APS/123-QED Constraining Extended Gravity Models by S2 star orbits around the Galactic Centre ∗ S. Capozziello,1,2,3, D. Borka,4 P. Jovanovi´c,5 and V. Borka Jovanovi´c4 1Dipartimento di Fisica, Universita` di Napoli ”Federico II”, Compl. Univ. di Monte S. Angelo, Ediﬁcio G, Via Cinthia, I-80126, Napoli, Italy, 2Istituto Nazionale di Fisica Nucleare (INFN) Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Ediﬁcio G, Via Cinthia, I-80126, Napoli, Italy, 3Gran Sasso Science Institute (INFN), Viale F. Crispi, 7, I-67100, L’Aquila, Italy. 4Atomic Physics Laboratory (040), Vinˇca Institute of Nuclear Sciences, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia. 5Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia. (Dated: May 26, 2020) We investigate the possibility to explain theoretically the observed deviations of S2 star orbit around the Galactic Centre using gravitational potentials derived from modiﬁed gravity models in absence of dark matter. To this aim, an analytic fourth-order theory of gravity, non-minimally coupled with a massive scalar ﬁeld is considered. Speciﬁcally, the interaction term is given by analytic functions f(R) and f(R, φ) where R is the Ricci scalar and φ is a scalar ﬁeld whose meaning can be related to further gravitational degrees of freedom. We simulate the orbit of S2 star around the Galactic Centre in f(R) (Yukawa-like) and f(R, φ) (Sanders-like) gravity potentials and compare it with NTT/VLT observations. Our simulations result in strong constraints on the range of gravity interaction. In the case of analytic functions f(R), we are not able to obtain reliable constraints on the derivative constants f and f , because the current observations of S2 star indicated that they 1 2 may be highly mutually correlated. In the case of analytic functions f(R, φ), we are able to obtain reliable constraints on the derivative constants f , f , f , f , f and f . The approach we 0 R RR φ φφ φR are proposing seems to be suﬃciently reliable to constrain the modiﬁed gravity models from stellar orbits around Galactic Centre. PACS numbers: 04.50.Kd, 04.25.Nx, 04.40.Nr I. INTRODUCTION generalizations of the Einstein theory where the gravita- tional action (the Hilbert-Einstein action) is assumed to be linear in the Ricci curvature scalar R. f(R) gravity Extended Theories of Gravity [1] are alternative theo- is a type of modiﬁed gravity which generalizes Einstein’s ries of gravitational interaction developed from the exact General Relativity and it was ﬁrst proposed in 1970 by starting points investigated ﬁrst by Einstein and Hilbert Buchdahl [12]. It is actually a family of models, each and aimed from one side to extend the positive results one deﬁned by a diﬀerent function of the Ricci scalar. of General Relativity and, on the other hand, to cure The simplest case is just the General Relativity. In the its shortcomings. Besides other fundamental issues, like case of f(R) gravity, one assumes a generic function f of dark energy and quantum gravity, these theories have the Ricci scalar R (in particular, analytic functions) and been proposed like alternative approaches to Newtonian searches for a theory of gravity having suitable behavior gravity in order to explain galactic and extragalactic dy- at small and large scale lengths. As a consequence of in- namics without introducing dark matter [2, 3]. In par- troducing an arbitrary function, there may be freedom to ticular, the search for non-Newtonian gravity is part of explain the accelerated expansion and structure forma- the quest for non-Einsteinian physics which consists of tion of the Universe without adding unknown forms of searching for deviations from Special and General Rela- dark energy or dark matter. One type of the Extended tivity [4–6]. They are aimed to address conceptual and Theories of Gravity is characterized by power-law La- experimental problems recently emerged in astrophysics grangians [13, 14]. Alternative approaches to Newtonian and cosmology from the observations of the Solar sys- gravity in the framework of the weak ﬁeld limit of fourth tem, binary pulsars, spiral galaxies, clusters of galaxies order gravity theory have been proposed and constraints and the large-scale structure of the Universe [7–11]. In on these theories have been discussed [15–22]. general, these theories describe gravity as a metric the- Yukawa-like corrections have been obtained in the ory with a linear connection but there are also aﬃne, or framework of f(R) gravity as a general feature of these metric-aﬃne formulations of Extended Theories of Grav- theories [23–26]. It is important to stress that they ity [1]. Essentially, they are based on straightforward emerge as exact solutions in the context of Extended Gravity and are not just put by hand as phenomeno- logical terms. The physical meaning of such corrections ∗ needs to be conﬁrmed at diﬀerent scales: for short dis- Corresponding author:\markcomm{footercolor}{capozziello@ na.infn.it} tances, Solar system, spiral galaxies and galaxy clusters. ##LTLine##

text: Constraining

label: title

text: Extended

text: Gravity

text: Models

text: by

text: star

text: orbits

text: around

text: the

text: Galactic

text: Centre

text: S.

label: author

text: D.

text: P.

text: and

text: V.

text: Borka

text: di

text: Fisica,

text: Universita`

text: di

text: Napoli

text: ”Federico

text: II”,

text: Compl.

text: Univ.

text: di

text: Monte

text: S.

text: Angelo,

text: Ediﬁcio

text: G,

text: Via

text: Cinthia,

text: Napoli,

text: Italy,

text: Nazionale

text: di

text: Fisica

text: Nucleare

text: (INFN)

text: Sez.

text: di

text: Napoli,

text: Compl.

text: Univ.

text: di

text: Monte

text: S.

text: Angelo,

text: Ediﬁcio

text: G,

text: Via

text: Cinthia,

text: Napoli,

text: Italy,

text: Sasso

text: Science

text: Institute

text: (INFN),

text: Viale

text: F.

text: Crispi,

text: L’Aquila,

text: Italy.

text: Physics

text: Laboratory

text: Vinˇca

text: Institute

text: of

text: Nuclear

text: Sciences,

text: University

text: of

text: Belgrade,

text: P.O.

text: Box

text: Belgrade,

text: Serbia.

text: Observatory,

text: Volgina

text: Belgrade,

text: Serbia.

text: (Dated:

text: May

text: We

text: investigate

text: the

text: possibility

text: to

text: explain

text: theoretically

text: the

text: observed

text: deviations

text: of

text: star

text: orbit

text: around

text: the

text: Galactic

text: Centre

text: using

text: gravitational

text: potentials

text: derived

text: from

text: modiﬁed

text: gravity

text: models

text: in

text: absence

text: of

text: dark

text: matter.

text: To

text: this

text: aim,

text: an

text: analytic

text: fourth-order

text: theory

text: of

text: gravity,

text: non-minimally

text: coupled

text: with

text: a

text: massive

text: scalar

text: ﬁeld

text: is

text: considered.

text: Speciﬁcally,

text: the

text: interaction

text: term

text: is

text: given

text: by

text: analytic

text: functions

text: f(R)

text: and

text: f(R,

text: φ)

text: where

text: R

text: is

text: the

text: Ricci

text: scalar

text: and

text: φ

text: is

text: a

text: scalar

text: ﬁeld

text: whose

text: meaning

text: can

text: be

text: related

text: to

text: further

text: gravitational

text: degrees

text: of

text: freedom.

text: We

text: simulate

text: the

text: orbit

text: of

text: star

text: around

text: the

text: Galactic

text: Centre

text: in

text: f(R)

text: (Yukawa-like)

text: and

text: f(R,

text: φ)

text: (Sanders-like)

text: gravity

text: potentials

text: and

text: compare

text: it

text: with

text: NTT/VLT

text: observations.

text: Our

text: simulations

text: result

text: in

text: strong

text: constraints

text: on

text: the

text: range

text: of

text: gravity

text: interaction.

text: In

text: the

text: case

text: of

text: analytic

text: functions

text: f(R),

text: we

text: are

text: not

text: able

text: to

text: obtain

text: reliable

text: constraints

text: on

text: the

text: derivative

text: constants

text: f

text: and

text: f

text: because

text: the

text: current

text: observations

text: of

text: star

text: indicated

text: that

text: they

text: may

text: be

text: highly

text: mutually

text: correlated.

text: In

text: the

text: case

text: of

text: analytic

text: functions

text: f(R,

text: φ),

text: we

text: are

text: able

text: to

text: obtain

text: reliable

text: constraints

text: on

text: the

text: derivative

text: constants

text: f

text: f

text: f

text: f

text: f

text: and

text: f

text: The

text: approach

text: we

text: R

text: RR

text: φ

text: φφ

text: φR

text: are

text: proposing

text: seems

text: to

text: be

text: suﬃciently

text: reliable

text: to

text: constrain

text: the

text: modiﬁed

text: gravity

text: models

text: from

text: stellar

text: orbits

text: around

text: Galactic

text: Centre.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: generalizations

text: of

text: the

text: Einstein

text: theory

text: where

text: the

text: gravita-

text: tional

text: action

text: (the

text: Hilbert-Einstein

text: action)

text: is

text: assumed

text: to

text: be

text: linear

text: in

text: the

text: Ricci

text: curvature

text: scalar

text: R.

text: f(R)

text: gravity

text: Extended

text: Theories

text: of

text: Gravity

text: are

text: alternative

text: theo-

text: is

text: a

text: type

text: of

text: modiﬁed

text: gravity

text: which

text: generalizes

text: Einstein’s

text: ries

text: of

text: gravitational

text: interaction

text: developed

text: from

text: the

text: exact

text: General

text: Relativity

text: and

text: it

text: was

text: ﬁrst

text: proposed

text: in

text: by

text: starting

text: points

text: investigated

text: ﬁrst

text: by

text: Einstein

text: and

text: Hilbert

text: Buchdahl

text: It

text: is

text: actually

text: a

text: family

text: of

text: models,

text: each

text: and

text: aimed

text: from

text: one

text: side

text: to

text: extend

text: the

text: positive

text: results

text: one

text: deﬁned

text: by

text: a

text: diﬀerent

text: function

text: of

text: the

text: Ricci

text: scalar.

text: of

text: General

text: Relativity

text: and,

text: on

text: the

text: other

text: hand,

text: to

text: cure

text: The

text: simplest

text: case

text: is

text: just

text: the

text: General

text: Relativity.

text: In

text: the

text: its

text: shortcomings.

text: Besides

text: other

text: fundamental

text: issues,

text: like

text: case

text: of

text: f(R)

text: gravity,

text: one

text: assumes

text: a

text: generic

text: function

text: f

text: of

text: dark

text: energy

text: and

text: quantum

text: gravity,

text: these

text: theories

text: have

text: the

text: Ricci

text: scalar

text: R

text: (in

text: particular,

text: analytic

text: functions)

text: and

text: been

text: proposed

text: like

text: alternative

text: approaches

text: to

text: Newtonian

text: searches

text: for

text: a

text: theory

text: of

text: gravity

text: having

text: suitable

text: behavior

text: gravity

text: in

text: order

text: to

text: explain

text: galactic

text: and

text: extragalactic

text: dy-

text: at

text: small

text: and

text: large

text: scale

text: lengths.

text: As

text: a

text: consequence

text: of

text: in-

text: namics

text: without

text: introducing

text: dark

text: matter

text: In

text: par-

text: troducing

text: an

text: arbitrary

text: function,

text: there

text: may

text: be

text: freedom

text: to

text: ticular,

text: the

text: search

text: for

text: non-Newtonian

text: gravity

text: is

text: part

text: of

text: explain

text: the

text: accelerated

text: expansion

text: and

text: structure

text: forma-

text: the

text: quest

text: for

text: non-Einsteinian

text: physics

text: which

text: consists

text: of

text: tion

text: of

text: the

text: Universe

text: without

text: adding

text: unknown

text: forms

text: of

text: searching

text: for

text: deviations

text: from

text: Special

text: and

text: General

text: Rela-

text: dark

text: energy

text: or

text: dark

text: matter.

text: One

text: type

text: of

text: the

text: Extended

text: tivity

text: They

text: are

text: aimed

text: to

text: address

text: conceptual

text: and

text: Theories

text: of

text: Gravity

text: is

text: characterized

text: by

text: power-law

text: La-

text: experimental

text: problems

text: recently

text: emerged

text: in

text: astrophysics

text: grangians

text: Alternative

text: approaches

text: to

text: Newtonian

text: and

text: cosmology

text: from

text: the

text: observations

text: of

text: the

text: Solar

text: sys-

text: gravity

text: in

text: the

text: framework

text: of

text: the

text: weak

text: ﬁeld

text: limit

text: of

text: fourth

text: tem,

text: binary

text: pulsars,

text: spiral

text: galaxies,

text: clusters

text: of

text: galaxies

text: order

text: gravity

text: theory

text: have

text: been

text: proposed

text: and

text: constraints

text: and

text: the

text: large-scale

text: structure

text: of

text: the

text: Universe

text: In

text: on

text: these

text: theories

text: have

text: been

text: discussed

text: general,

text: these

text: theories

text: describe

text: gravity

text: as

text: a

text: metric

text: the-

text: Yukawa-like

text: corrections

text: have

text: been

text: obtained

text: in

text: the

text: ory

text: with

text: a

text: linear

text: connection

text: but

text: there

text: are

text: also

text: aﬃne,

text: or

text: framework

text: of

text: f(R)

text: gravity

text: as

text: a

text: general

text: feature

text: of

text: these

text: metric-aﬃne

text: formulations

text: of

text: Extended

text: Theories

text: of

text: Grav-

text: theories

text: It

text: is

text: important

text: to

text: stress

text: that

text: they

text: ity

text: Essentially,

text: they

text: are

text: based

text: on

text: straightforward

text: emerge

text: as

text: exact

text: solutions

text: in

text: the

text: context

text: of

text: Extended

text: Gravity

text: and

text: are

text: not

text: just

text: put

text: by

text: hand

text: as

text: phenomeno-

text: logical

text: terms.

text: The

text: physical

text: meaning

text: of

text: such

text: corrections

text: needs

text: to

text: be

text: conﬁrmed

text: at

text: diﬀerent

text: scales:

text: for

text: short

text: dis-

text: Corresponding

text: na.infn.it}

text: tances,

text: Solar

text: system,

text: spiral

text: galaxies

text: and

text: galaxy

text: clusters.

text: ##LTLine##


text: 2 A compilation of experimental, geophysical and astro- to stress that the current astrometric limit is not suﬃ- nomical constraints on Yukawa violations of the grav- cient to unambiguously conﬁrm such a claim. Weinberg itational inverse square law are given in Figs. 9 and et al. [42] discussed physical experiments achievable via 10 from [27] for diﬀerent ranges. These results show the monitoring of stellar dynamics near the Galactic Cen- that the Yukawa term is relatively well constrained for tre with a diﬀraction-limited, next-generation, extremely the short ranges. For longer distances Yukawa correc- large telescope (ELT). tions have been successfully applied to clusters of galax- The aim of this paper is to give the astronomical con- ies [23, 28, 29]. Lucchesi and Peron [30] analyzed peri- straints on Extended Theories of Gravity by using the pe- center general relativistic precession and gave constraints culiar dynamics of S2 star. In particular, we want to ﬁx on exponential potential to Solar System measurements. the ranges of Yukawa-like correction parameters adopt- However, further tests are needed in order to set robust ing the NTT/VLT observations. The paper is organized constraints on Yukawa corrections. Galactic stellar dy- as follows. Sec. II is devoted to a short summary of namics could be of great aid in this program. Extended Gravity in view of the Newtonian limit where S-stars are mainly young early-type stars that closely Yukawa-like corrections emerge. The simulated orbits of orbit the massive compact object at the center of Milky S2 star in modiﬁed potential are considered in Sec. III. ∗ Way, named Sgr A [31–36]. These stars, together with In particular, we set the problem of how to constrain the recently discovered dense gas cloud falling towards the Yukawa-potential parameters. Sec. IV is devoted to the Galactic Centre [37], indicate that the massive central results of the simulation. Conclusions are drawn in Sec. object is a black hole. In our simulation we will treat V. central object like ”massive compact object” since our II. EXTENDED THEORIES OF GRAVITY goal was only to study orbits of stars around Galactic Centre, no matter what is the nature of the object (black Examples of Extended Theories of Gravity are higher- hole or not). For at least one of them, called S2, there are order, scalar-tensor gravities, see, for example [1, 3, 10, some observational indications that its orbit maybe devi- 43–49]. These theories can be characterized by two ates from the Keplerian case due to relativistic precession main features: i) the geometry can non-minimally cou- [33, 38]. ple to some scalar ﬁeld; ii) higher-order curvature in- However, we have to point out that the present astro- variants can appear into the action. In the ﬁrst case, metric limit is still not suﬃcient to deﬁnitely conﬁrm such we are dealing with scalar-tensor gravity, and in the sec- a claim. On the other hand, the astrometric accuracy is ond case we have higher-order gravity. Combinations of constantly improving from around 10 mas during the ﬁrst non-minimally coupled and higher order terms can also part of the observational period, currently reaching less emerge in eﬀective Lagrangian, producing mixed higher than 1 mas (0.3 mas) see [39]. Furthermore, some recent order/scalar-tensor gravity. A general class of higher- studies provide more and more evidence that the orbit order-scalar-tensor theories in four dimensions is given of S2 star is not closing (see e.g. Fig. 2 in [38]). Here, by the eﬀec(cid:90)tive action(cid:2)[1, 50]: we ﬁtted the NTT/VLT astrometric observations of S2 √ star, which contain a possible indication for orbital pre- S − (cid:3) (cid:3) (cid:3) cession around the massive compact object at Galactic = d4x g f(R, R, 2R, . . . , kR, φ)+ Centre, in order to constrain the parameters of Sanders- XL ω(φ)φ φ;α + ] , (1) like gravity potential, since this kind of potential has not ;α m been tested at these scales yet. We obtained much larger orbital precession of S2 star in Sanders-like gravity than where f is an unspeciﬁed function of curvature invari- the corresponding value predicted by General Relativity. X ants and scalar ﬁeld φ and = 8πG. Here we use the In the paper [33] page 1092, Fig. 13, authors presented convention c = 1. the Keplerian orbit but they have to move the position of The simplest extension of General Relativity is central point mass to explain orbital precession. In our achieved assuming: orbit, calculated by Sanders-like potential for best ﬁtting parameters, we also obtained precession, but with a ﬁxed → position of the central point mass. In other words, we do R f(R) , ω(φ) = 0 (2) not need to move central point mass in order to get the ﬁt. As a general remark, the orbit of S2 will give as- where the actio(cid:90)n (1) becomes [50]: tronomers the opportunity to test for various eﬀects pre- dicted by General Relativity. The orbital precession can √ S − XL occur due to relativistic eﬀects, resulting in a prograde = d4x g [f(R) + ] . (3) m pericentre shift or due to a possible extended mass dis- tribution, producing a retrograde shift [40]. Both pro- grade relativistic and retrograde Newtonian pericentre A general gravitational potential, with a Yukawa cor- shifts will result in rosette shaped orbits [41]. We have rection, can be obtained in the Newtonian limit of any ##LTLine## ##LTLine##

text: A

text: compilation

text: of

text: experimental,

text: geophysical

text: and

text: astro-

text: to

text: stress

text: that

text: the

text: current

text: astrometric

text: limit

text: is

text: not

text: suﬃ-

text: nomical

text: constraints

text: on

text: Yukawa

text: violations

text: of

text: the

text: grav-

text: cient

text: to

text: unambiguously

text: conﬁrm

text: such

text: a

text: claim.

text: Weinberg

text: itational

text: inverse

text: square

text: law

text: are

text: given

text: in

text: Figs.

text: and

text: et

text: al.

text: discussed

text: physical

text: experiments

text: achievable

text: via

text: from

text: for

text: diﬀerent

text: ranges.

text: These

text: results

text: show

text: the

text: monitoring

text: of

text: stellar

text: dynamics

text: near

text: the

text: Galactic

text: Cen-

text: that

text: the

text: Yukawa

text: term

text: is

text: relatively

text: well

text: constrained

text: for

text: tre

text: with

text: a

text: diﬀraction-limited,

text: next-generation,

text: extremely

text: the

text: short

text: ranges.

text: For

text: longer

text: distances

text: Yukawa

text: correc-

text: large

text: telescope

text: (ELT).

text: tions

text: have

text: been

text: successfully

text: applied

text: to

text: clusters

text: of

text: galax-

text: The

text: aim

text: of

text: this

text: paper

text: is

text: to

text: give

text: the

text: astronomical

text: con-

text: ies

text: Lucchesi

text: and

text: Peron

text: analyzed

text: peri-

text: straints

text: on

text: Extended

text: Theories

text: of

text: Gravity

text: by

text: using

text: the

text: pe-

text: center

text: general

text: relativistic

text: precession

text: and

text: gave

text: constraints

text: culiar

text: dynamics

text: of

text: star.

text: In

text: particular,

text: we

text: want

text: to

text: ﬁx

text: on

text: exponential

text: potential

text: to

text: Solar

text: System

text: measurements.

text: the

text: ranges

text: of

text: Yukawa-like

text: correction

text: parameters

text: adopt-

text: However,

text: further

text: tests

text: are

text: needed

text: in

text: order

text: to

text: set

text: robust

text: ing

text: the

text: NTT/VLT

text: observations.

text: The

text: paper

text: is

text: organized

text: constraints

text: on

text: Yukawa

text: corrections.

text: Galactic

text: stellar

text: dy-

text: as

text: follows.

text: Sec.

text: II

text: is

text: devoted

text: to

text: a

text: short

text: summary

text: of

text: namics

text: could

text: be

text: of

text: great

text: aid

text: in

text: this

text: program.

text: Extended

text: Gravity

text: in

text: view

text: of

text: the

text: Newtonian

text: limit

text: where

text: S-stars

text: are

text: mainly

text: young

text: early-type

text: stars

text: that

text: closely

text: Yukawa-like

text: corrections

text: emerge.

text: The

text: simulated

text: orbits

text: of

text: orbit

text: the

text: massive

text: compact

text: object

text: at

text: the

text: center

text: of

text: Milky

text: star

text: in

text: modiﬁed

text: potential

text: are

text: considered

text: in

text: Sec.

text: III.

text: Way,

text: named

text: Sgr

text: A

text: These

text: stars,

text: together

text: with

text: In

text: particular,

text: we

text: set

text: the

text: problem

text: of

text: how

text: to

text: constrain

text: the

text: recently

text: discovered

text: dense

text: gas

text: cloud

text: falling

text: towards

text: the

text: Yukawa-potential

text: parameters.

text: Sec.

text: IV

text: is

text: devoted

text: to

text: the

text: Galactic

text: Centre

text: indicate

text: that

text: the

text: massive

text: central

text: results

text: of

text: the

text: simulation.

text: Conclusions

text: are

text: drawn

text: in

text: Sec.

text: object

text: is

text: a

text: black

text: hole.

text: In

text: our

text: simulation

text: we

text: will

text: treat

text: V.

text: central

text: object

text: like

text: ”massive

text: compact

text: object”

text: since

text: our

text: II.

text: EXTENDED

text: THEORIES

text: OF

text: GRAVITY

text: goal

text: was

text: only

text: to

text: study

text: orbits

text: of

text: stars

text: around

text: Galactic

text: Centre,

text: no

text: matter

text: what

text: is

text: the

text: nature

text: of

text: the

text: object

text: (black

text: Examples

text: of

text: Extended

text: Theories

text: of

text: Gravity

text: are

text: higher-

text: hole

text: or

text: not).

text: For

text: at

text: least

text: one

text: of

text: them,

text: called

text: there

text: are

text: order,

text: scalar-tensor

text: gravities,

text: see,

text: for

text: example

text: some

text: observational

text: indications

text: that

text: its

text: orbit

text: maybe

text: devi-

text: These

text: theories

text: can

text: be

text: characterized

text: by

text: two

text: ates

text: from

text: the

text: Keplerian

text: case

text: due

text: to

text: relativistic

text: precession

text: main

text: features:

text: i)

text: the

text: geometry

text: can

text: non-minimally

text: cou-

text: ple

text: to

text: some

text: scalar

text: ﬁeld;

text: ii)

text: higher-order

text: curvature

text: in-

text: However,

text: we

text: have

text: to

text: point

text: out

text: that

text: the

text: present

text: astro-

text: variants

text: can

text: appear

text: into

text: the

text: action.

text: In

text: the

text: ﬁrst

text: case,

text: metric

text: limit

text: is

text: still

text: not

text: suﬃcient

text: to

text: deﬁnitely

text: conﬁrm

text: such

text: we

text: are

text: dealing

text: with

text: scalar-tensor

text: gravity,

text: and

text: in

text: the

text: sec-

text: a

text: claim.

text: On

text: the

text: other

text: hand,

text: the

text: astrometric

text: accuracy

text: is

text: ond

text: case

text: we

text: have

text: higher-order

text: gravity.

text: Combinations

text: of

text: constantly

text: improving

text: from

text: around

text: mas

text: during

text: the

text: ﬁrst

text: non-minimally

text: coupled

text: and

text: higher

text: order

text: terms

text: can

text: also

text: part

text: of

text: the

text: observational

text: period,

text: currently

text: reaching

text: less

text: emerge

text: in

text: eﬀective

text: Lagrangian,

text: producing

text: mixed

text: higher

text: than

text: mas

text: mas)

text: see

text: Furthermore,

text: some

text: recent

text: order/scalar-tensor

text: gravity.

text: A

text: general

text: class

text: of

text: higher-

text: studies

text: provide

text: more

text: and

text: more

text: evidence

text: that

text: the

text: orbit

text: order-scalar-tensor

text: theories

text: in

text: four

text: dimensions

text: is

text: given

text: of

text: star

text: is

text: not

text: closing

text: (see

text: e.g.

text: Fig.

text: in

text: Here,

text: by

text: the

text: we

text: ﬁtted

text: the

text: NTT/VLT

text: astrometric

text: observations

text: of

text: star,

text: which

text: contain

text: a

text: possible

text: indication

text: for

text: orbital

text: pre-

text: S

text: cession

text: around

text: the

text: massive

text: compact

text: object

text: at

text: Galactic

text: g

text: f(R,

text: R,

text: kR,

text: φ)+

text: Centre,

text: in

text: order

text: to

text: constrain

text: the

text: parameters

text: of

text: Sanders-

text: XL

text: ω(φ)φ

text: φ;α

text: like

text: gravity

text: potential,

text: since

text: this

text: kind

text: of

text: potential

text: has

text: not

text: ;α

text: m

text: been

text: tested

text: at

text: these

text: scales

text: yet.

text: We

text: obtained

text: much

text: larger

text: orbital

text: precession

text: of

text: star

text: in

text: Sanders-like

text: gravity

text: than

text: where

text: f

text: is

text: an

text: unspeciﬁed

text: function

text: of

text: curvature

text: invari-

text: the

text: corresponding

text: value

text: predicted

text: by

text: General

text: Relativity.

text: X

text: ants

text: and

text: scalar

text: ﬁeld

text: φ

text: and

text: Here

text: we

text: use

text: the

text: In

text: the

text: paper

text: page

text: Fig.

text: authors

text: presented

text: convention

text: c

text: the

text: Keplerian

text: orbit

text: but

text: they

text: have

text: to

text: move

text: the

text: position

text: of

text: The

text: simplest

text: extension

text: of

text: General

text: Relativity

text: is

text: central

text: point

text: mass

text: to

text: explain

text: orbital

text: precession.

text: In

text: our

text: achieved

text: assuming:

text: orbit,

text: calculated

text: by

text: Sanders-like

text: potential

text: for

text: best

text: ﬁtting

text: parameters,

text: we

text: also

text: obtained

text: precession,

text: but

text: with

text: a

text: ﬁxed

text: position

text: of

text: the

text: central

text: point

text: mass.

text: In

text: other

text: words,

text: we

text: do

text: R

text: f(R)

text: ω(φ)

text: not

text: need

text: to

text: move

text: central

text: point

text: mass

text: in

text: order

text: to

text: get

text: the

text: ﬁt.

text: As

text: a

text: general

text: remark,

text: the

text: orbit

text: of

text: will

text: give

text: as-

text: where

text: the

text: becomes

text: tronomers

text: the

text: opportunity

text: to

text: test

text: for

text: various

text: eﬀects

text: pre-

text: dicted

text: by

text: General

text: Relativity.

text: The

text: orbital

text: precession

text: can

text: S

text: XL

text: occur

text: due

text: to

text: relativistic

text: eﬀects,

text: resulting

text: in

text: a

text: prograde

text: g

text: [f(R)

text: m

text: pericentre

text: shift

text: or

text: due

text: to

text: a

text: possible

text: extended

text: mass

text: dis-

text: tribution,

text: producing

text: a

text: retrograde

text: shift

text: Both

text: pro-

text: grade

text: relativistic

text: and

text: retrograde

text: Newtonian

text: pericentre

text: A

text: general

text: gravitational

text: potential,

text: with

text: a

text: Yukawa

text: cor-

text: shifts

text: will

text: result

text: in

text: rosette

text: shaped

text: orbits

text: We

text: have

text: rection,

text: can

text: be

text: obtained

text: in

text: the

text: Newtonian

text: limit

text: of

text: any

text: ##LTLine##

text: ##LTLine##


text: 7 0.00018 and m = -0.0026, respectively. is much smaller than the value of precession per orbital φ The simulated orbits of S2 star around the Galactic period in Sanders gravity potential, but the direction of Centre in Sanders gravity potential (blue solid line) and the precession is the same. in Newtonian gravity potential (red dashed line) for α V. CONCLUSIONS = 0.00018 and m = -0.0026 during 10 periods, are pre- φ sented in Fig. 4. We can notice that the precession of S2 In this paper, we compared the observed and simulated star orbit has the same direction as in General Relativity. S2 star orbits around the Galactic Centre, in order to con- The precession of S2 star orbit in the same direction can strain the parameters of gravitational potentials derived be also obtained for some ranges of parameter δ in gen- from modiﬁed gravity models. The obtained results are eral Yukawa potential (for more details see paper [21]). quite comfortable for the eﬀective gravitational poten- As it can be read from Fig. 4, the best ﬁ◦t orbit in Sanders tial derived from f(R, φ) gravity that, essentially, repro- gravity potential precesses for about 3 .1 per orbital pe- duce Sanders-like potentials [56, 58] phenomenologically riod. adopted to explain the rotation curves of spiral galaxies. In case of Sanders potential, analytical calculation of Also if such kind of potentials are not suﬃcient in ad- orbital precession is very complicated to obtain, so we dressing completely the problem of dark matter in galax- calculated it numerically and presented in Figs. 5 and 6 ies [59], they give indications that alternative theories of as a function of α and mφ. Assuming that a potential gravity could be viable in describing galactic dynamics. does not diﬀer signiﬁcantly from Newtonian potential, we In other words, orbital solutions derived from such a derive perturbing potential from: potential are in good agreement with the reduced χ2 de- duced for Keplerian orbits. This fact allows to ﬁx the − −GM range of variation for α and mφ, the two parameters V (r) = Φ (r) ΦN (r) ; ΦN (r) = r . (13) characterizing the potential (10). The precession of S2 star orbit obtained for the best ﬁt parameter values (α = 0.00018 and m = -0.0026) has the positive direction, Obtained perturbing potent(cid:16)ial is of the form: (cid:17) φ as in General Relativity. In particular, we ﬁtted the NTT/VLT astrometric ob- √ − GMα − − · · − servations of S2 star, which contain a possible indication V (r) = e 1 3α mφ r 1 , (14) r(1 + α) for orbital precession around massive compact object at Galactic centre, in order to constrain the parameters of Sanders-like gravity potential, since this theory has not and it can be used for calculating the precession angle been tested at these scales yet. We obtained much larger according to the equation (30) from paper [41]: (cid:90) orbital precession of S2 star in Sanders-like gravity than the corresponding value predicted by General Relativity. − 1 · In the paper [34], the authors presented Newtonian orbit 2L √z dz dV (z) ∆θ = − , (15) but with moved position of central point mass in diﬀer- GMe2 1 z2 dz ent way. In that way they explained observed precession. − 1 In our calculated orbit by Sanders like potential for best ﬁtting parameters, we also obtained precession but with L where r is related to z via: r = . By diﬀerenti- ﬁxed position of central point mass, since we have not to (cid:0) (cid:1) 1 + ez move it. ating the perturbing potential V (z) and substituting its derivative and expression for the semilatus rectum of the However, one should keep in mind that we considered − orbital ellipse (L = a 1 e2 ) in above equation (15), an idealized model ignoring many uncertainty factors, and taking same values for orbital elements of S2 star such as extended mass distributions, perturbations from like in paper [19] we obtained numerically for α = 0.00018 non-symmetric mass distribution, etc. Therefore, future and m = -0.0026 that precession per orbital period is observations with advanced facilities, such as GRAVITY ◦ φ 3 .053. which will enable extrem∼ely accurate measurements of Graphical presentation of precession per orbital pe- the positions of stars ∼of 10 µas [60], or E-ELT with riod for α in the range [0.0001, 0.0003] and m in expected accuracy of 50-100 µas [61], are needed in − − φ [ 0.003, 0.002] is given in Fig. 5, and the case for α order to verify the claims in this paper. As ﬁnal remark, − − − in [ 0.0005, 0.0005] and m in [ 0.003, 0.0025] is pre- we believe that the surveys aimed to give details on the φ sented in Fig. 6. As one can see pericenter advance (like dynamics around the Galactic Centre could be a powerful in GR) is obtained for positive α, and retrograde pre- tool to test theories of gravity. cession for negative α. However, it should be taken into Acknowledgments account that ﬁts better than Keplerian are obtained only for positive α and hence for the precession in the same S.C. acknowledges the support of INFN (iniziativa direction as in GR. speciﬁca TEONGRAV). D.B., P.J. and V.B.J. wish to General Relativity predicts that pericenter of S2 star acknowledge the support by the Ministry of Education, ◦ should advance by 0 .08 per orbital revolution [34] which Science and Technological Development of the Republic ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: m

text: respectively.

text: is

text: much

text: smaller

text: than

text: the

text: value

text: of

text: precession

text: per

text: orbital

text: φ

text: The

text: simulated

text: orbits

text: of

text: star

text: around

text: the

text: Galactic

text: period

text: in

text: Sanders

text: gravity

text: potential,

text: but

text: the

text: direction

text: of

text: Centre

text: in

text: Sanders

text: gravity

text: potential

text: (blue

text: solid

text: line)

text: and

text: the

text: precession

text: is

text: the

text: same.

text: in

text: Newtonian

text: gravity

text: potential

text: (red

text: dashed

text: line)

text: for

text: α

text: V.

label: section

text: CONCLUSIONS

text: and

text: m

text: during

text: periods,

text: are

text: pre-

text: φ

text: sented

text: in

text: Fig.

text: We

text: can

text: notice

text: that

text: the

text: precession

text: of

text: In

text: this

text: paper,

text: we

text: compared

text: the

text: observed

text: and

text: simulated

text: star

text: orbit

text: has

text: the

text: same

text: direction

text: as

text: in

text: General

text: Relativity.

text: star

text: orbits

text: around

text: the

text: Galactic

text: Centre,

text: in

text: order

text: to

text: con-

text: The

text: precession

text: of

text: star

text: orbit

text: in

text: the

text: same

text: direction

text: can

text: strain

text: the

text: parameters

text: of

text: gravitational

text: potentials

text: derived

text: be

text: also

text: obtained

text: for

text: some

text: ranges

text: of

text: parameter

text: δ

text: in

text: gen-

text: from

text: modiﬁed

text: gravity

text: models.

text: The

text: obtained

text: results

text: are

text: eral

text: Yukawa

text: potential

text: (for

text: more

text: details

text: see

text: paper

text: quite

text: comfortable

text: for

text: the

text: eﬀective

text: gravitational

text: poten-

text: As

text: it

text: can

text: be

text: read

text: from

text: Fig.

text: the

text: best

text: ﬁ◦t

text: orbit

text: in

text: Sanders

text: tial

text: derived

text: from

text: f(R,

text: φ)

text: gravity

text: that,

text: essentially,

text: repro-

text: gravity

text: potential

text: precesses

text: for

text: about

text: per

text: orbital

text: pe-

text: duce

text: Sanders-like

text: potentials

text: phenomenologically

text: riod.

text: adopted

text: to

text: explain

text: the

text: rotation

text: curves

text: of

text: spiral

text: galaxies.

text: In

text: case

text: of

text: Sanders

text: potential,

text: analytical

text: calculation

text: of

text: Also

text: if

text: such

text: kind

text: of

text: potentials

text: are

text: not

text: suﬃcient

text: in

text: ad-

text: orbital

text: precession

text: is

text: very

text: complicated

text: to

text: obtain,

text: so

text: we

text: dressing

text: completely

text: the

text: problem

text: of

text: dark

text: matter

text: in

text: galax-

text: calculated

text: it

text: numerically

text: and

text: presented

text: in

text: Figs.

text: and

text: ies

text: they

text: give

text: indications

text: that

text: alternative

text: theories

text: of

text: as

text: a

text: function

text: of

text: α

text: and

text: mφ.

text: Assuming

text: that

text: a

text: potential

text: gravity

text: could

text: be

text: viable

text: in

text: describing

text: galactic

text: dynamics.

text: does

text: not

text: diﬀer

text: signiﬁcantly

text: from

text: Newtonian

text: potential,

text: we

text: In

text: other

text: words,

text: orbital

text: solutions

text: derived

text: from

text: such

text: a

text: derive

text: perturbing

text: potential

text: from:

text: potential

text: are

text: in

text: good

text: agreement

text: with

text: the

text: reduced

text: χ2

text: de-

text: duced

text: for

text: Keplerian

text: orbits.

text: This

text: fact

text: allows

text: to

text: ﬁx

text: the

text: −GM

text: range

text: of

text: variation

text: for

text: α

text: and

text: mφ,

text: the

text: two

text: parameters

text: V

text: (r)

text: Φ

text: (r)

text: ΦN

text: (r)

text: ΦN

text: (r)

text: r

text: characterizing

text: the

text: potential

text: The

text: precession

text: of

text: star

text: orbit

text: obtained

text: for

text: the

text: best

text: ﬁt

text: parameter

text: values

text: (α

text: and

text: m

text: has

text: the

text: positive

text: direction,

text: Obtained

text: perturbing

text: is

text: of

text: the

text: form:

text: φ

text: as

text: in

text: General

text: Relativity.

text: In

text: particular,

text: we

text: ﬁtted

text: the

text: NTT/VLT

text: astrometric

text: ob-

text: GMα

text: servations

text: of

text: star,

text: which

text: contain

text: a

text: possible

text: indication

text: V

text: (r)

text: e

text: 3α

text: mφ

text: r

text: α)

text: for

text: orbital

text: precession

text: around

text: massive

text: compact

text: object

text: at

text: Galactic

text: centre,

text: in

text: order

text: to

text: constrain

text: the

text: parameters

text: of

text: Sanders-like

text: gravity

text: potential,

text: since

text: this

text: theory

text: has

text: not

text: and

text: it

text: can

text: be

text: used

text: for

text: calculating

text: the

text: precession

text: angle

text: been

text: tested

text: at

text: these

text: scales

text: yet.

text: We

text: obtained

text: much

text: larger

text: according

text: to

text: the

text: equation

text: from

text: paper

text: orbital

text: precession

text: of

text: star

text: in

text: Sanders-like

text: gravity

text: than

text: the

text: corresponding

text: value

text: predicted

text: by

text: General

text: Relativity.

text: In

text: the

text: paper

text: the

text: authors

text: presented

text: Newtonian

text: orbit

text: √z

text: dz

text: dV

text: (z)

text: ∆θ

text: but

text: with

text: moved

text: position

text: of

text: central

text: point

text: mass

text: in

text: diﬀer-

text: dz

text: ent

text: way.

text: In

text: that

text: way

text: they

text: explained

text: observed

text: precession.

text: In

text: our

text: calculated

text: orbit

text: by

text: Sanders

text: like

text: potential

text: for

text: best

text: ﬁtting

text: parameters,

text: we

text: also

text: obtained

text: precession

text: but

text: with

text: L

text: where

text: r

text: is

text: related

text: to

text: z

text: via:

text: r

text: By

text: diﬀerenti-

text: ﬁxed

text: position

text: of

text: central

text: point

text: mass,

text: since

text: we

text: have

text: not

text: to

text: ez

text: move

text: it.

text: ating

text: the

text: perturbing

text: potential

text: V

text: (z)

text: and

text: substituting

text: its

text: derivative

text: and

text: expression

text: for

text: the

text: semilatus

text: rectum

text: of

text: the

text: However,

text: one

text: should

text: keep

text: in

text: mind

text: that

text: we

text: considered

text: orbital

text: ellipse

text: (L

text: a

text: in

text: above

text: equation

text: an

text: idealized

text: model

text: ignoring

text: many

text: uncertainty

text: factors,

text: and

text: taking

text: same

text: values

text: for

text: orbital

text: elements

text: of

text: star

text: such

text: as

text: extended

text: mass

text: distributions,

text: perturbations

text: from

text: like

text: in

text: paper

text: we

text: obtained

text: numerically

text: for

text: α

text: non-symmetric

text: mass

text: distribution,

text: etc.

text: Therefore,

text: future

text: and

text: m

text: that

text: precession

text: per

text: orbital

text: period

text: is

text: observations

text: with

text: advanced

text: facilities,

text: such

text: as

text: GRAVITY

text: φ

text: which

text: will

text: enable

text: extrem∼ely

text: accurate

text: measurements

text: of

text: Graphical

text: presentation

text: of

text: precession

text: per

text: orbital

text: pe-

text: the

text: positions

text: of

text: stars

text: ∼of

text: µas

text: or

text: E-ELT

text: with

text: riod

text: for

text: α

text: in

text: the

text: range

text: and

text: m

text: in

text: expected

text: accuracy

text: of

text: µas

text: are

text: needed

text: in

text: φ

text: is

text: given

text: in

text: Fig.

text: and

text: the

text: case

text: for

text: α

text: order

text: to

text: verify

text: the

text: claims

text: in

text: this

text: paper.

text: As

text: ﬁnal

text: remark,

text: in

text: and

text: m

text: in

text: is

text: pre-

text: we

text: believe

text: that

text: the

text: surveys

text: aimed

text: to

text: give

text: details

text: on

text: the

text: φ

text: sented

text: in

text: Fig.

text: As

text: one

text: can

text: see

text: pericenter

text: advance

text: (like

text: dynamics

text: around

text: the

text: Galactic

text: Centre

text: could

text: be

text: a

text: powerful

text: in

text: GR)

text: is

text: obtained

text: for

text: positive

text: α,

text: and

text: retrograde

text: pre-

text: tool

text: to

text: test

text: theories

text: of

text: gravity.

text: cession

text: for

text: negative

text: α.

text: However,

text: it

text: should

text: be

text: taken

text: into

text: Acknowledgments

text: account

text: that

text: ﬁts

text: better

text: than

text: Keplerian

text: are

text: obtained

text: only

text: for

text: positive

text: α

text: and

text: hence

text: for

text: the

text: precession

text: in

text: the

text: same

text: S.C.

text: acknowledges

text: the

text: support

text: of

text: INFN

text: (iniziativa

text: direction

text: as

text: in

text: GR.

text: speciﬁca

text: TEONGRAV).

text: D.B.,

text: P.J.

text: and

text: V.B.J.

text: wish

text: to

text: General

text: Relativity

text: predicts

text: that

text: pericenter

text: of

text: star

text: acknowledge

text: the

text: support

text: by

text: the

text: Ministry

text: of

text: Education,

text: should

text: advance

text: by

text: per

text: orbital

text: revolution

text: which

text: Science

text: and

text: Technological

text: Development

text: of

text: the

text: Republic

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: PARTIAL AND TOTAL IDEALS OF VON NEUMANN ALGEBRAS NADISH DE SILVA RUI SOARES BARBOSA Abstract. A notion of partial ideal for an operator algebra is a weakening the notion of ideal where the deﬁning algebraic condi- tions are enforced only in the commutative subalgebras. We show that, in a von Neumann algebra, the ultraweakly closed two-sided ideals, which we call total ideals, correspond to the unitarily invari- ant partial ideals. The result also admits an equivalent formulation in terms of central projections. We place this result in the context of an investigation into notions of spectrum of noncommutative ∗ C -algebras. 1. Introduction The principal theorem proved in this paper concerns the von Neu- ∗ mann algebraic analogue of a conjecture made for C -algebras. We ∗ describe the original question concerning C -algebras, state the von Neumann algebraic version, and outline the more general investigation into noncommutative topology in the context of which the original question ﬁrst arose. In the next sections, we prove the principal the- orem after establishing some technical preliminaries. We conclude by ∗ indicating some ideas about how to tackle the C -algebraic case. All algebras and subalgebras considered throughout this paper are ∗ A assumed to be unital. By a total ideal of a C -algebra , we mean a A norm closed, two-sided ideal of . ∗ A Deﬁnition 1.1. A partial ideal of a C -algebra is a map π that ∗ A assigns to each commutative sub-C -algebra V of a closed ideal of V (cid:48) ∩ ⊂ (cid:48) such that π(V ) = π(V ) V whenever V V .

text: PARTIAL

text: AND

text: TOTAL

text: IDEALS

text: OF

text: VON

text: NEUMANN

text: ALGEBRAS

text: NADISH

text: DE

text: SILVA

text: RUI

text: SOARES

text: BARBOSA

text: Abstract.

text: A

label: abstract

text: notion

text: of

text: partial

text: ideal

text: for

text: an

text: operator

text: algebra

text: is

text: a

text: weakening

text: the

text: notion

text: of

text: ideal

text: where

text: the

text: deﬁning

text: algebraic

text: condi-

text: tions

text: are

text: enforced

text: only

text: in

text: the

text: commutative

text: subalgebras.

text: We

text: show

text: that,

text: in

text: a

text: von

text: Neumann

text: algebra,

text: the

text: ultraweakly

text: closed

text: two-sided

text: ideals,

text: which

text: we

text: call

text: total

text: ideals,

text: correspond

text: to

text: the

text: unitarily

text: invari-

text: ant

text: partial

text: ideals.

text: The

text: result

text: also

text: admits

text: an

text: equivalent

text: formulation

text: in

text: terms

text: of

text: central

text: projections.

text: We

text: place

text: this

text: result

text: in

text: the

text: context

text: of

text: an

text: investigation

text: into

text: notions

text: of

text: spectrum

text: of

text: noncommutative

text: C

text: -algebras.

text: Introduction

text: The

text: principal

text: theorem

text: proved

text: in

text: this

text: paper

text: concerns

text: the

text: von

text: Neu-

text: mann

text: algebraic

text: analogue

text: of

text: a

text: conjecture

text: made

text: for

text: C

text: -algebras.

text: We

text: describe

text: the

text: original

text: question

text: concerning

text: C

text: -algebras,

text: state

text: the

text: von

text: Neumann

text: algebraic

text: version,

text: and

text: outline

text: the

text: more

text: general

text: investigation

text: into

text: noncommutative

text: topology

text: in

text: the

text: context

text: of

text: which

text: the

text: original

text: question

text: ﬁrst

text: arose.

text: In

text: the

text: next

text: sections,

text: we

text: prove

text: the

text: principal

text: the-

text: orem

text: after

text: establishing

text: some

text: technical

text: preliminaries.

text: We

text: conclude

text: by

text: indicating

text: some

text: ideas

text: about

text: how

text: to

text: tackle

text: the

text: C

text: -algebraic

text: case.

text: All

text: algebras

text: and

text: subalgebras

text: considered

text: throughout

text: this

text: paper

text: are

text: A

text: assumed

text: to

text: be

text: unital.

text: By

text: a

text: total

text: ideal

text: of

text: a

text: C

text: -algebra

text: we

text: mean

text: a

text: A

text: norm

text: closed,

text: two-sided

text: ideal

text: of

text: A

text: Deﬁnition

text: A

text: partial

text: ideal

text: of

text: a

text: C

text: -algebra

text: is

text: a

text: map

text: π

text: that

text: A

text: assigns

text: to

text: each

text: commutative

text: sub-C

text: -algebra

text: V

text: of

text: a

text: closed

text: ideal

text: of

text: V

text: such

text: that

text: π(V

text: π(V

text: V

text: whenever

text: V

text: V