TITLE: $Z_{2}$- graded structures for $C_{red} ^{*} (F_{2})$
QUESTION [5 upvotes]: Let $F_{2}$ be the free group with two generators. 
Then $F_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z_{2}$ graded structure for $C^{*}_{red} (F_{2})$, in a natural way.
My question:

Is there another $Z_{2}$ graded structure for $C^{*}_{red} (F_{2})$  which is not graded isomorphic to the above $Z_{2}$- graded structure?

Please consider the same question for $C^{*}_{red} (F_{1}) \sim C(\mathbb{S}^{1})$.
Namely: is there a $Z_{2}$ graded structure for $C(\mathbb{S}^{1})$ which is  not graded isomorphism to the standard grading structure(decomposition to even and odd continuous functions)?

REPLY [6 votes]: It is well-known that $K_1(C^*_{\rm red}(F_2))={\mathbb Z^2}$ with generators given by $[u]$ and $[v]$, where $F_2=\langle u,v\rangle$.
Now, the automorphism of order two associated with the even-odd grading is multiplying each generator by $-1$, which is homotopic to the multiplication by $1$. Hence it is trivial on $K_1$.
On the other side, the automorphism of order two that comes mapping $u$ to $v$ and $v$ to $u$ gives rise to a non-trivial action on $K_1$. Thus, the two automorphisms of order two cannot be conjugate - and hence the associated graded algebras cannot be isomorphic.