TITLE: Universal $C^*$-algebra with generators and relations
QUESTION [14 upvotes]: We say that the $C^*$-algebra $A$ generated by $a_1,...,a_n$ is universal subject to relations $R_1,...,R_m$ if for every $C^*$-algebra $B$ with elements $b_1,...,b_n$ satisfying relations $R_1,...,R_m$ there is $C^*$-epimorphism $\varphi: A \to B$ such that $\varphi(a_i)=b_i$. One of the basic examples is the $C^*$-algebra of complex valued function on three sphere $C(S^3)$ which is the universal commutative unital $C^*$-algebra generated by $a,b$ with relation $a^*a+b^*b=1$. My question is the following: what kind of relations can we impose on our $C^*$-algebra? In all examples which I saw the relations were algebraic and were of the form: $f(a_1,...,a_n,a_1^*,...,a_n^*)=0$ where $f$ was some polynomial. In particular do we admit:

quantification and referring to other elements not being the generators
order properties of $C^*$-algebras
functions which are no longer polynomials (continuous functions, Borel functions etc.)

If the answer is positive I would be grateful to know some (known in literature) examples of universal $C^*$-algebras arising in such a way.

REPLY [18 votes]: Since free C*-algebras don't exist, we can't give a concrete description of all relations that are allowed.  Instead, we need to give axioms that determine what collections of n-tuples $(a_1,\dots,a_n)$ in $A$ are allowed, where $A$ varies over all C*-algebras.  It is important that the elements not "know about" the ambient C*-algebra, so ``a is in a separable C*-algebra'' is not allowed.
Some of what is allowed is conditions like $0 \leq x \leq 1$ and 
$$
0\leq\left[\begin{array}{cc}
\mathbf{1} & x\\
x^{*} & \mathbf{1}
\end{array}\right]\leq 1
$$
where $\mathbf{1}$ is in the unitization of $A$.  Another fun example is ``x is hermitian and has spectrum contained in the Cantor set''.  We can't use Borel functional calculus, but we can take a continuous function on $\mathbb R$ that is bounded and use as a relation "$x$ is normal and $f(x)=0$."  See section 5 of "From Matrix to Operator Inequalities" by me, Canad. Math. Bull. 55(2012), 339--350 for a use of analytic functional calculus.
I could drone on forever here, but I did already: ``C*-Algebra Relations'' in Mathematica Scandinavica 107, 43--72, 2010.