Idea and Intuition

Let \mathcal{G}=\{g_i\}_{i\geq 1} be a (usually finite) set of generators and \mathcal{R} a (usually finite) set of relations between the generators. The generators at this point are indeterminates, and we will be momentarily vague about what is and isn’t a relation. We write (if it exists!) \mathrm{C}^*(\mathcal{G}\mid \mathcal{R}) for the universal \mathrm{C}^*-algebra generated by generators \mathcal{G} and relations \mathcal{R}.

It has the following universal property. Suppose |\mathcal{G}|=n. Let A be a \mathrm{C}^*-algebra with elements f_1,\dots,f_n that satisfy the relations \mathcal{R}, then there is a (unique) *-homomorphism \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to A mapping g_i\mapsto f_i. This map will be a surjective *-homomorphism \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to \mathrm{C}^*(f_1,\dots,f_n) (aka a quotient map and \mathrm{C}^*(f_1,\dots,f_n) a quotient of \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})).

If the f_i\in A generate A, \mathrm{C}^*(f_1,\dots,f_n), then \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to A is a surjective *-homomorphism.

My (highly non-rigourous, the tilde reminding of hand-waving) intuition for this object is that you collect all (really all, not just isomorphism classes, we want below C([0,1]) and e.g. C([0,2]) of \mathrm{C}^*-algebras A_\lambda generated by n generators satisfying the relations \mathcal{R} and forming a “big direct sum/product thingy” out of all of them:

\displaystyle \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\sim \widetilde{\prod_{\lambda\in \Lambda}}A_\lambda.

Then the *-homomorphism \pi_\mu:A\to A_\mu given by the universal property is given by projection onto that factor (which is a surjective *-homomorphism, a quotient):

\displaystyle\pi_\mu:\widetilde{\prod_{\lambda\in \Lambda}}A_\lambda\to A_\mu.

This intuition works well but we should give a brief account of things are done properly.

First off, not every relation will give a universal \mathrm{C}^*-algebra. For example, consider \mathcal{G}=\{g\} and \mathcal{R}=\{g=g^*\}. The problem here is one of norm. Recall our rough intuition from above. What is the norm on a\in \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}), this big thing \widetilde{\prod_{\lambda\in\Lambda}}A_\lambda? It should be something like, where the norm of A_\lambda is \|\cdot\|_\lambda the supremum over the factors:

\displaystyle \|f\|\sim\sup_{\lambda\in \Lambda}\|\pi_{\lambda}(f)\|_{\lambda}\qquad (f\in \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})).

The first approach to show that \mathrm{C}^*(g\mid g=g^*) does not exist is to consider for each \alpha\in [0,\infty) the \mathrm{C}^*-algebra A_\alpha=C([0,\alpha]) which is singly-generated by the self-adjoint f_\alpha:=t\mapsto t. But the norm \|\cdot \|_\alpha in A_\alpha is the supremum norm, so we find \|f_\alpha\|_{\alpha}=\alpha. From here:

\displaystyle \|g\|\sim\sup_{\lambda\in \Lambda}\|\pi_{\lambda}(g)\|_{\lambda}\geq \sup_{\alpha\in [0,\infty)}\|f_{\alpha}\|_{\alpha}=\sup_{\alpha\in [0,\infty)}\alpha,

which is unbounded. The relations must give a bounded norm to the generators.

As an example of relations that do bound the generators, consider, n self-adjoint generators such that the sum of their squares is the identity:

\displaystyle \mathrm{C}^*\left(g_1,\dots,g_n\mid g_i=g_i^*,\,\sum_i g_i^2=1\right).

These relations bounds the norm of the generators \|g_i\|\leq 11, and this gives existence to this algebra, using the Gelfand philosophy giving the “algebra of continuous functions on the free sphere”, C(\mathcal{S}^{n-1}_+) (I think first considered by Banica and Goswami).

Note here the relations are given by polynomial relations. If the polynomial relations are suitably admissible (i.e. give a bound to the generators), in this setting there is a real construction (real vs our ridiculous \widetilde{\prod}) of \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}). See p.885 (link to *.pdf quantum group lecture notes of Moritz Weber).

In fact, this is only a small class of the possible relations. I suggest there are at least two more types:

  • \mathrm{C}^* relations that would be (admissible) norm relations (for example, in one generator, adding \|g\|\leq 1, a non-polynomial relation, to the polynomial relation g=g^* gives an admissible set of relations, and \mathrm{C}^*(g\mid g=g^*,\|g\|\leq 1)\cong C([-1,1]). For \mathrm{C}^*/norm relations see here and maybe here.
  • (admissible) strong relations (see here for a use of this, with reference)

The constructions in one or both of cases might be constructive, as in the case (admissible) polynomial relations, but there is also an approach using category theory. But the main feature in all such definitions is the universal property, whose use could be summarised as follows:

Let \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}) be a universal \mathrm{C}^*-algebra. The universal property can be used to answer questions about \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}) such as:

  • is some polynomial p(g_1,\dots,g_n) in the generators non-zero,
  • is \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}) infinite dimensional,
  • is \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}) non-commutative;

because, if A is a \mathrm{C}^*-algebra with elements f_1,\dots,f_n that satisfy the relations then there is a unique *-homomorphism \pi:\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to \mathrm{C}^*(f_1,\dots,f_n). So, for example, where B is some such \mathrm{C}^*(f_1,\dots,f_n) then

  • if p(f_1,\dots,f_n)\neq 0, then p(g_1,\dots,g_n)\neq 0 as \pi(p(g_1,\dots,g_n))=p(f_1,\dots,f_n),
  • if B is infinite dimensional then \pi is a surjective *-homomorphism onto an infinite dimensional algebra, and so the domain \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}) is infinite dimensional too.
  • if the commutator [f_i,f_j]\neq 0, so that B is non-commutative, then so is \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}) as \pi([g_i,g_j])=[f_i,f_j]

These quotients B can be considered models of \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}).

Two Examples

A projection p in a \mathrm{C}^*-algebra is such that p=p^2=p^*. Consider

A:=\mathrm{C}^*(p,q,1\mid p\text{ and }q\text{ are projections}).

Existence is easy, because the norm of a non-zero projection is one. To answer questions about this algebra consider the infinite dihedral group D_\infty=\langle a,b\mid a^2=b^2=e\rangle. This has group algebra \mathbb{C}D_\infty and group \mathrm{C}^*-algebra \mathrm{C}^*(D_\infty). Note that a and b in \mathrm{C}^*(D_\infty) satisfy the relations of A, and so we have a *-homomorphism (in fact a *-isomorphism) \pi:A\to \mathrm{C}^*(D_\infty). This tells us that any monomial in the generators of A is non-zero, A is infinite dimensional, and A is non-commutative.

A partition of unity is a finite set of projections that sum to the identity, \sum_{i}p_i=1. A magic unitary in a \mathrm{C}^*-algebra A is a matrix u\in M_N(A) such that the entries along any one row or column form a partition of unity. Consider (notation to be kept mysterious):

\displaystyle C(S_N^+)=\mathrm{C}^*\left(u_{ij}\mid u\text{ an }N\times N\text{ magic unitary}\right).

Consider the following magic unitary:

\displaystyle v=\begin{bmatrix}p & 1-p & 0 & 0 \\ 1-p & p & 0 & 0 \\ 0 & 0 & q & 1-q \\ 0 & 0 & 1-q & q\end{bmatrix}.

Note that the v_{ij} satisfy the relations of C(S_N^+) and in fact generate A=\mathrm{C}^*(p,q) from above. Thus we have a quotient \pi:C(S_4^+)\to A which shows that C(S_4^+) is infinite dimensional and noncommutative.

It is possible to show using a magic unitary with entries in M_N(\mathbb{C}) that for N>3, a monomial with entries in u\in M_N(C(S_N^+)) is zero for trivial reasons only (link to *.pdf, from Theorem 1 on).

In addition it can be shown that for u (and similarly v) the matrix in M_N(C(S_N^+)\otimes_{\min} C(S_N^+)) with (i,j)– entries

\displaystyle \sum_{k=1}^Nu_{ik}\otimes u_{kj}

is a magic unitary, and thus by the universal property u_{ij}\mapsto \sum_{k=1}^Nu_{ik}\otimes u_{kj} is a *-homomorphism… the comultiplication giving C(S_N^+) the structure of a compact quantum group.

Commutative Examples

If a universal \mathrm{C}^* algebra is commutative (as in commutativity, [g_i,g_j]=0, is one of the relations, vs the relations imply commutativity, as in C(S_3^+) (nice exercise)), we write \mathrm{C}^*_{\text{comm}}(\mathrm{G}\mid \mathrm{R}). In this case Gelfand’s Theorem, that \mathrm{C}^*_{\text{comm}}(\mathrm{G}\mid \mathrm{R})\cong C(\text{characters}), often allows us to easily identity the algebra (vs the noncommutative case where the universal algebra is mostly studied via models, quotients).

Theorem

If A is a (polynomial) universal commutative \mathrm{C}^*-algebra, then it of the form C(X), and X is given by the tuples (z_1,\dots,z_n)\subset\mathbb{C}^n that satisfy the relations of of A.

Proof: Characters are *-homomorphisms A\to \mathbb{C}.

Suppose that z_1,\dots,z_n satisfy the relations. Then by the universal property, \varphi(g_i)=z_i is a *-homomorphism.

On the contrary, suppose that \varphi is a character. Then the relations are preserved under a *-homomorphism.

Examples

For A_0:=\mathrm{C}_{\text{comm}}^*(p,q,1\mid p\text{ and }q\text{ are projections}), projections in \mathbb{C} are just the scalars zero and one. Thus the spectrum is \{(0,0),(1,0),(0,1),(1,1)\} and A_0 is the algebra of continuous functions on four points.

For \mathrm{C}_{\text{comm}}^*\left(u_{ij}\mid u\text{ an }N\times N\text{ magic unitary}\right) collect the tuple of N^2 generators in a matrix. The relations imply that each such tuple is in fact a permutation matrix, and so the universal algebra above is the algebra of continuous functions on S_N.

For

\displaystyle \mathrm{C}_{\text{comm}}^*\left(g_1,\dots,g_n\mid g_i=g_i^*,\,\sum_i g_i^2=1\right)

you end up with tuples of n real numbers in [-1,1] whose sum of squares is one… otherwise known as the sphere \mathcal{S}^{n-1}.

Liberations

An interesting business here is to start with a universal commutative \mathrm{C}^* algebra, say one of the three examples above… and see do you get something strictly bigger, necessarily non-commutative, if you drop commutativity. In the above, yes you do. Gelfand’s theorem says that a commutative unital \mathrm{C}^*-algebra is the algebra of continuous functions on a compact space X (which we call a classical space). The Gelfand Philosophy says therefore that a noncommutative unital \mathrm{C}^*-algebra A can be thought of as the algebra of continuous functions on a compact quantum space \mathbb{X}. Note here \mathbb{X} is not a set-of-points, but a virtual object, and strictly A=C(\mathbb{X}) is just notation (but see here).

Liberating the second example above from commutativity is the passage from the permutation group S_N to the quantum permutation group S_N^+. Liberating the third example above gives the passage from the real sphere \mathcal{S}^{n-1} to a quantum sphere called the free sphere \mathcal{S}^{n-1}_+.

We can also, of course, work in the other direction, imposing commutativity on not-necessarily universal \mathcal{A}. If we write \mathcal{A}=C(\mathbb{X}), then imposing commutativity (qotienting by commutator ideal) gives us the classical version X of \mathbb{X}.

Imposing commutativity is not so scary: using the above you just get \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to C(\text{characters})… and everything we said above about identifying characters on \mathrm{C}_{\text{comm}}^*(\mathcal{G}\mid\mathcal{R}) holds for \mathrm{C}^*(\mathcal{G}\mid\mathcal{R}) too.

This can be used: for example if a quantum group acts on a structure S, then its classical version acts on S. This idea was used by Banica and I to show that not every quantum permutation group is the quantum automorphism group of a finite graph (link to *.pdf).

  1. If you have positive elements f_i in a \mathrm{C}^*-algebra with bounded sum, say \|\sum_i f_i\|\leq C then we can bound the summands \|f_i\|\leq C. Write \sum_if_i=f. Note f is positive with norm C. Let \varphi be a state such that \varphi(f_j)=\|f_j\| and apply this to f, \varphi(\sum_i f_i)=\varphi(f) which yields, with \varphi bounded of norm one, \varphi(f_j)+\sum_{i\neq j}\varphi(f_i)\leq C. The result follows by positivity of the sum and the state. To apply to the above play with the \mathrm{C}^* identity. Incidentally, I cannot remember how Murphy proves the existence of \varphi but I like for positive f in a \mathrm{C}^*-algebra, \mathrm{C}^*(f)\cong C(\sigma(f)). Then let x:=\max_{\lambda\in\sigma(f)}\lambda and define a state \mathrm{ev}_x on \mathrm{C}^*(f)\cong C(\sigma(f)). It is the case that \mathrm{ev}_x(f)=f(x)=\|f\|_{C(\sigma(f))}=x, also equal to the norm of f in the ambient algebra (by a spectral radius theorem). Extend the evaluation functional to the whole algebra by Hahn-Banach. ↩︎