Noncommutative spaces

The central dogma of geometry is that a space X is the “same” data as the functions on X, i.e. the functions X \to R for R some ring. In analysis, of course, we usually take R to be the real numbers, or an algebra over the real numbers. Some examples of this phenomenon:

  • A smooth manifold X is determined up to smooth diffeomorphism by the sheaf of rings of smooth functions from open subsets of X into \mathbb R.
  • An open subset X of the Riemann sphere is determined up to conformal transformation is determined by the algebra of holomorphic functions X \to \mathbb C.
  • An algebraic variety X over a field K is determined up to isomorphism by the sheaf of rings of polynomial functions from Zariski open subsets of X into K.
  • The example we will focus on in this post: a compact Hausdorff space X is determined by the C*-algebra of continuous functions X \to \mathbb C.

In the above we have always assumed that the ring R was commutative, and in fact a field. As a consequence the algebra A of functions X \to R is also commutative. If we view A as “the same data” as X, then there should be some generalization of the notion of a space that corresponds to when A is noncommutative. I learned how to do this for compact Hausdorff spaces when I took a course on C*-algebras last semester, and I’d like to describe this generalization of the notion of compact Hausdorff space here.

Fix a Hilbert space H, and let B(H) be the algebra of bounded linear operators on H. By a C*-algebra acting on H, we mean a complex subalgebra of B(H) which is closed under taking adjoints and closed with respect to the operator norm. In case H = L^2(X) for some measure space X, we will refer to C*-algebras acting on H as C*-algebras on X.

In case X is a compact Hausdorff space, there are three natural C*-algebras on X. First is B(H) itself; second is the algebra B_0(H) of compact operators acting on H (in other words, the norm-closure of finite rank operators), and third is the C*-algebra C(X) of continuous functions on X, which acts on H by pointwise multiplication. The algebra C(X) is of course commutative.

If A is a commutative, unital C*-algebra and I is a maximal ideal in A, then A/I is a field, and by the Gelfand-Mazur theorem in fact A/I = \mathbb C. It follows that the maximal spectrum (i.e. the set of maximal ideals) of A is in natural bijection with the space of continuous, surjective algebra morphisms A \to \mathbb C; namely, I corresponds to the projection A \to A/I. In particular I is closed. One can show that every such morphism has norm 1, so lies in the unit ball of the dual space A*; and that the limit of a net of morphisms in the weakstar topology is also a morphism. Therefore the weakstar topology of A* restricts to the maximal spectrum of A, which is then a compact Hausdorff space by the Banach-Alaoglu theorem.

If A = C(X), then the maximal spectrum of A consists of the ideals I_x of functions f such that f(x) = 0, where x \in X. The projections A \to A/I_x are exactly of the form f \mapsto f(x). This is the content of the baby Gelfand-Naimark theorem: the maximal spectrum of C(X) is X, and conversely every commutative, unital C*-algebra arises this way. (The great Gelfand-Naimark theorem, on the other hand, guarantees that every C*-algebra, defined as a special kind of ring rather than analytically, is a C*-algebra acting [faithfully] on a Hilbert space as I defined above.) The baby Gelfand-Naimark theorem is far from constructive; the proof of the Banach-Alaoglu theorem requires the axiom of choice, especially when A is not separable.

Let us briefly run through the properties of X that we can easily recover from C(X). Continuous maps X \to Y correspond to continuous, unital algebra morphisms C(Y) \to C(X); a map f is sent to the morphism which pulls back functions along f. Points, as noted above, correspond to maximal ideals. If K is a closed subset of X and I is the ideal of functions that vanish on K, then A/I is the “localization” of A at I. (So inclusions of compact Hausdorff spaces correspond to projections of C*-algebras.)

If X is just locally compact, then the C*-algebra C_0(X) of continuous functions on X which vanish at the fringe of every compactification of X is not unital (since 1 does not vanish at the fringe), and the unital algebras A which extend C_0(X) correspond to compact Hausdorff spaces that contain X, in a sort of “Galois correspondence”. In fact, the one-point compactification of X corresponds to the minimal unital C*-algebra containing C_0(X), while the Stone-Cech compactification corresponds to the C*-algebra C_b(X) of bounded continuous functions on X. Therefore the compactifications of X correspond to the unital C*-algebras A such that C_0(X) \subset A \subseteq C_b(X), and surjective continuous functions which preserve the compactification structure, say Z \to Y, correspond to the inclusion maps C(Y) \to C(Z). Two examples of this phenomenon:

  • The C*-algebra C_b(\mathbb N) = \ell^\infty has a maximal spectrum whose points are exactly the ultrafilters on \mathbb N. In particular \ell^\infty/c_0 has a maximal spectrum whose points are exactly the free ultrafilters. This is why we needed the axiom of choice so badly.
  • The compactification of \mathbb R obtained as the C*-algebra generated by C_0(\mathbb R) and $\theta \mapsto e^{i\theta}$ is a funny-looking curve in \mathbb R^3. Try drawing it! (As a hint: think of certain spaces which are connected but not path-connected…)

If X = (X, d) is a metric space, then we can recover the metric d from its Lipschitz seminorm L. This is the seminorm Lf = \sup_{x_1, x_2} |f(x_1)-f(x_2)|/d(x_1,x_2), which is finite exactly if f is Lipschitz. One then has d(x_1, x_2) = \sup_{Lf \leq 1} |f(x_1) - f(x_2)|. The Lipschitz seminorm also satisfies the identities L1 = 0, and L(fg) \leq ||f||_\infty Lg + ||g||_\infty Lf, the latter being known as the Leibniz axiom. A seminorm \rho satisfying these properties gives rise to a metric on X, and if the resulting topology on X is actually the topology of X, we say that \rho is a Lip-norm.

Finally, if E \to X is a continuous vector bundle, then let \Gamma(E) denote the vector space of continuous sections of E. Then \Gamma(E) is a projective module over C(X), and Swan’s theorem says that every projective module arises this way. Moreover, the module structure of \Gamma(E) determines E up to isomorphism.

So now we have all we need to consider noncommutative compact Hausdorff spaces. By such a thing, I just mean a unital C*-algebra A, which I will call X when I am thinking of it as a space. Since A is noncommutative, we cannot appeal to the Gelfand-Mazur theorem, and in fact the notion of a maximal ideal doesn’t quite make sense. The correct generalization of maximal ideal is known as a “primitive ideal”: an ideal I is primitive if I is the annihilator of a simple A-module. (The only simple A-module is \mathbb C if A is commutative, so in that case I is maximal.) The primitive spectrum of A admits a Zariski topology, which coincides with the weakstar topology when A is commutative. So the points of X will consist of primitive ideals of A. Metrics on X will be Lip-norms; vector bundles will be projective modules.

What of functions? We already know that elements of A should be functions on X. But what is their codomain? Clearly not a field — they aren’t commutative! If I is a primitive ideal corresponding to a point x, we let A/I be the localization at x. This will be some C*-algebra, which is a simple module A_x that we view as the ring that functions send x into. (So this construction may send different points into different rings — this isn’t so different from algebraic geometry, however, where localization at an integer n sends points of \text{Spec} \mathbb Z into the ring \mathbb Z/n.) Matrix rings are simple, so often A_x will be a matrix ring.

Let’s run through an example of a noncommutative space. Let T be the unit circle in $\mathbb R^2$. The group $\mathbb Z/2$ acts on T by (x, y) \mapsto (x, -y). This corresponds to an action on the C*-algebra C(T) by f(x, y) \mapsto f(x, -y). Whenever a group G acts on a C*-algebra B, we can define a semidirect product B \rtimes G, and so we let our C*-algebra A be the semidirect product C(T) \rtimes \mathbb Z/2. It turns out that A is the C*-algebra generated by unitary operators that form a group isomorphic to \mathbb Z/2 * \mathbb Z/2, where * is the coproduct of groups.

We may express elements of A as f + gb, where b is the nontrivial element of \mathbb Z/2 and f,g \in C(T); A is a C*-algebra acting on L^2(T) by (f + gb)\xi(x, y) = f(x, y)\xi(x, y) + g(x, y)\xi(x, -y), for any \xi \in L^2(T). The center Z(A) of A, therefore, consists of f \in C(T) such that f(x, y) = f(x, -y). Therefore Z(A) is isomorphic to the C*-algebra C([-1, 1]), where f \in Z(A) is sent to \tilde f(x) = f(x, \pm y) (where x^2 + y^2 = 1, and the choice of sign does not matter by assumption on f).

The spectrum of Z(A) is easy to compute, therefore: it is just [-1, 1]! For every x \in [-1, 1], let A_x = A/I_x be “localization at x”, where I_x is the ideal generated by f \in Z(A) which vanish at x. Now let \cos \theta = x; one then has a morphism of C*-algebras A_x \to \mathbb C^{2 \times 2} by $f + gb \mapsto \frac{1}{2}\begin{bmatrix}f(e^{i\theta})&g(e^{i\theta})\\g(e^{-i\theta})&f(e^{-i\theta})\end{bmatrix}$. This morphism is always injective, and if x \in (-1, 1), it is also surjective. Therefore A_x = \mathbb C^{2 \times 2} in that case. Since \mathbb C^{2 \times 2} is a simple ring, it follows that the spectrum of A contains x.

But something funny happens at the end-points, corresponding to the fact that (\pm 1, 0) were fixed points of \mathbb Z/2. Since f(e^{i\theta}) = f(e^{-i\theta}) in that case and similarly for g, A_x is isomorphic to the 2-dimensional subalgebra of \mathbb C^{2 \times 2} consisting of symmetric matrices with a doubled diagonal entry. This is not a simple module, and in fact projects onto \mathbb C in two different ways; therefore there are two points in the spectrum of A corresponding to each of \pm 1!

Thus the primitive ideal space X of A is not Hausdorff; in fact it looks like [-1, 1], except that the end-points are doubled. This is similar to the “bug-eyed line” phenomenon in algebraic geometry.

What is the use of such a space? Well, Qiaochu Yuan describes a proof (that I believe is due to Marc Rieffel), using this space X, that if B is any C*-algebra and p, q \in B are projections such that ||p - q|| < 1, then there is a unitary operator u such that pu = uq at MathOverflow. The idea is that p, q actually project onto generators of the C*-algebra A = C(T) \rtimes \mathbb Z/2, using the fact that A is also generated by \mathbb Z/2 * \mathbb Z/2.

As a consequence, in any separable C*-algebra, there are only countably many projections. Thus one may view the above discussion as a very overcomplicated, long-winded proof of the fact that B(\ell^2) is not separable.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s