Crossed products of irrational rotation algebras by finite subgroups of SL 2 (Z) are AF

Crossed products of irrational rotation algebras by finite subgroups of SL 2 (Z) are AF N. Christopher Phillips 7 May 2008 N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 1 / 36

The Sixth Annual Spring Institute on Noncommutative Geometry and Operator Algebras 5 14 May 2008 Vanderbilt University, Department of Mathematics, Nashville, Tennessee, USA. This is joint work with Siegfried Echterhoff, Wolfgang Lück, and Samuel Walters. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 2 / 36

The main result of this talk is now several years old. I am presenting it here because it gives a connection between the Elliott classification program and the work surrounding the Baum-Connes conjecture: both play major roles in the proof. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 3 / 36

More recent results Theorem (with Weaver) The Continuum Hypothesis implies that the Calkin algebra L(H)/K(H) has outer automorphisms. Theorem (Farah) It is consistent with ZFC that the Calkin algebra has no outer automorphisms. These have little connection with either noncommutative geometry or the Elliott classification program. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 4 / 36

Rotation algebras Let θ R. The rotation algebra A θ is the universal C*-algebra generated by unitaries u and v satisfying vu = exp(2πiθ)uv. For θ = 0, it is C(S 1 S 1 ), the continuous functions on the ordinary torus. For this reason, the rotation algebras are often called noncommutative tori. For rational θ, say θ = p q in lowest terms, A θ is the section algebra of a locally trivial continuous field over S 1 S 1, with fiber M q. For θ R \ Q, the algebra A θ, now called an irrational rotation algebra, is simple and nuclear. Any two unitaries satisfying the relation vu = exp(2πiθ)uv generate a copy of A θ. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 5 / 36

Rotation algebras (continued) The irrational rotation algebras have long been among the most intensively studied examples of C*-algebras. Among other things, their smooth subalgebras were among the first examples studied in noncommutative geometry. There is no space here to describe the long history, but I want to mention one theorem: Theorem (Elliott-Evans) The irrational rotation algebras are simple AT algebras with real rank zero. AT algebras are a kind of direct limit algebra; see below. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 6 / 36

Action of SL 2 (Z) The group SL 2 (Z) acts on R 2, and thus on R 2 /Z 2 = S 1 S 1. This action generalizes to an action on the rotation algebra A θ, defined by sending the matrix ( ) n1,1 n n = 1,2 n 2,1 n 2,2 to the automorphism determined by and α n (u) = exp(πin 1,1 n 2,1 θ)u n 1,1 v n 2,1 α n (v) = exp(πin 1,2 n 2,2 θ)u n 1,2 v n 2,2. (Check that the elements on the right satisfy the same relations that u and v do.) N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 7 / 36

Finite subgroups of SL 2 (Z) Up to conjugacy, there are four nontrivial finite subgroups of SL 2 (Z). They are generated by ( ) ( ) ( ) ( ) 1 0 1 1 0 1 0 1,,, and, 0 1 1 0 1 0 1 1 and are isomorphic to Z 2, Z 3, Z 4, and Z 6. Each of these gives an action of the appropriate cyclic group on each A θ. In the formulas for the actions of these subgroups, one can omit the scalar factors with no essential change. Thus, for example, we take the generator of the action of Z 4 to send u to v and v to u. This particular action is sometimes called the noncommutative Fourier transform. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 8 / 36

The commutative case Consider θ = 0, so we are looking at actions of Z 2, Z 3, Z 4, and Z 6 on S 1 S 1. The noncommutative Fourier transform is rotation by π 2. All four orbit spaces turn out to be homeomorphic to S 2, but not smoothly: there is a small finite number of nonfree orbits, which give corners. The quotients are all orbifolds. The C*-algebra crossed product C (Z k, S 1 S 1 ) reflects this structure. It has the following form. There is a finite list of unital subalgebras of M k, and one gets the subalgebra of C(S 2, M k ) consisting of functions whose values at certain points (finitely many) are in these subalgebras. For example, for k = 2, one gets {f C(S 2, M 2 ): f (x 1 ), f (x 2 ) C C}, for two points x 1, x 2 S 2 and with C C identified with the diagonal matrices in M 2. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 9 / 36

The rational case Let θ = p q in lowest terms. The structures of both the crossed product C (Z k, A θ ) and the fixed point algebra A Z k θ have been worked out (Bratteli-Elliott-Evans-Kishimoto for k = 2 and Farsi-Watling for the other cases). The answers are very much like for C (Z k, S 1 S 1 ) (but with larger matrices), with a consistent pattern of finite dimensional subalgebras. Exception: For very small q, there is some simplification in the fixed point algebras. For example, C(S 1 S 1 ) Z k = C((S 1 S 1 )/Z k ) = C(S 2 ). N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 10 / 36

K-theory The Pimsner-Voiculescu exact sequence shows that (ignoring order) K (A θ ) is the same for all θ. Thus, it is the same as for C(S 1 S 1 ): K 0 (A θ ) = Z 2 and K 1 (A θ ) = Z 2. The computations described above allowed the computation of K (C (Z k, A θ )) for rational θ. The results are: and for all k. K 0 (C (Z 2, A θ )) = Z 6, K 0 (C (Z 3, A θ )) = Z 8, K 0 (C (Z 4, A θ )) = Z 9, K 0 (C (Z 6, A θ )) = Z 10, K 1 (C (Z k, A θ )) = 0 The fixed point subalgebras have the same K-theory, except for very small denominators. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 11 / 36

The conjecture Kumjian proved (after Bratteli-Elliott-Evans-Kishimoto, but long before Farsi-Watling) that the computation of K (C (Z 2, A θ )) is valid for all θ, not just for θ Q. The method was to realize this C*-algebra as a crossed product C (Z 2 Z 2, S 1 ). There is no analog for the crossed products by Z 3, Z 4, and Z 6. The K-theory computations suggested that K (C (Z k, A θ )) has the K-theory of an AF algebra (direct limit of finite dimensional C*-algebras; see below). For θ R \ Q, the crossed product C (Z k, A θ ) is simple, and the conjecture was made that it is in fact an AF algebra for all k and all irrational θ. (Actually, the conjecture was made for the fixed point algebras, but in this case the two versions are equivalent.) N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 12 / 36

Some results This was proved by Bratteli-Kishimoto in 1992 for k = 2, again using the realization as C (Z 2 Z 2, S 1 ). Much later Walters obtained some partial results for k = 4, but the problem remained open for a long time. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 13 / 36

The result Theorem (with Echterhoff, Lück, and Walters) C (Z k, A θ ) is AF for all k {2, 3, 4, 6} and all irrational θ. Moreover, the (unordered) K-theory is the same as for θ Q. It follows that the fixed point algebras A Z k θ are also AF. We can also compute the order on K 0, and thus determine exactly which AF algebras one gets. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 14 / 36

Crossed products Crossed products have already been discussed in a number of talks here. I just want to point out one thing: if G is finite and α is an action of G on a C*-algebra A, then the crossed product C (G, A, α) is just the skew group ring A[G], with a suitable adjoint and norm. No completion is needed. Nevertheless, crossed products by finite groups are often very hard to understand. For example, it is easier to compute the K-theory of crossed products by Z, by F n, and even by R, than it is to compute the K-theory of crossed products by Z 2. (There exists a contractible C*-algebra A, so that K (A) = 0, and an action α: Z 2 Aut(A), such that K (C (Z, A, α)) 0.) N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 15 / 36

AF algebras AF algebras have been talked about less. Definition An AF algebra is a C*-algebra obtained as a countable direct limit of finite dimensional C*-algebras. (Recall that every finite dimensional C*-algebra is a finite direct sum of finite full matrix algebras.) Since K-theory commutes with direct limits, it follows that if A is AF, then K 1 (A) = 0 and K 0 (A) is torsion free. Theorem (Bratteli) Let A be a separable C*-algebra. Then A is AF if and only if for every finite set S A and every ε > 0, there exists a finite dimensional subalgebra F A such that dist(a, F ) < ε for all a S. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 16 / 36

AT algebras The following was used in the statement of the Elliott-Evans Theorem: Definition An AT algebra is a C*-algebra obtained as a countable direct limit of finite direct sums of C*-algebras of the form M n, C([0, 1], M n ), and C(S 1, M n ). If A is an AT algebra, then K 0 (A) and K 1 (A) are torsion free. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 17 / 36

Outline of the proof We outline the proof that C (Z k, A θ ) is AF. One might hope to write down an explicit direct system of finite dimensional C*-algebras, and prove that the direct limit is isomorphic to C (Z k, A θ ). No such proof is known for k 2. Instead the proof proceeds via two steps: 1 Compute K (C (Z k, A θ )). 2 Prove that C (Z k, A θ ) is classifiable. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 18 / 36

Computing K (C (Z k, A θ )) We describe the first step. The aim is to show that K (C (Z k, A θ )) is independent of θ, so that it suffices to compute K (C (Z k, A θ )) when θ = 0. This case is already known, or can be computed from the Baum-Connes conjecture. The method uses group C*-algebras twisted by cocycles, and the Baum-Connes conjecture. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 19 / 36

Cocycles Let G be a discrete group. A (circle valued 2-) cocycle on G is a function ω : G G S 1 such that ω(h, k)ω(g, hk) = ω(g, h)ω(gh, k) and ω(1, g) = ω(g, 1) = 1 for g, h, k G. (In general, one should allow G to be just locally compact, and ask that ω be a Borel function. For simplicity, we stick to the discrete case.) These cocycles are part of the cohomology of groups (in the discrete case; Borel cohomology when the group is locally compact). For a cocycle ω on G, there is a twisted group C*-algebra C (G, ω). It is generated by unitaries u g for g G, satisfying u g u h = ω(g, h)u gh. (This condition defines an ω-representation of G.) There is also a reduced C*-algebra C r (G, ω), which is the image of C (G, ω) under a twisted version of the regular representation. In all specific cases here, we will have C r (G, ω) = C (G, ω). N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 20 / 36

Rotation algebras via cocycles Take G = Z 2, and let θ R. Then A θ = C (Z 2, ω) for ω θ ((m 1, m 2 ), (n 1, n 2 )) = exp(πiθ(m 2 n 1 m 1 n 2 )). The point is that sending the standard generators of Z 2 to u, v A θ gives an ω θ -representation of Z 2 in A θ. Now take G = Z 2 SL 2 (Z). One checks that ω is invariant under the action of SL 2 (Z) on Z 2, and it follows that there is a canonical extension of ω θ to G, given by ((m, g), (n, h)) ω θ (m, g(n)) for m, n Z 2 and g, h SL 2 (Z). If we restrict to the finite subgroup Z k SL 2 (Z), the resulting twisted group C*-algebra turns out to C (Z k, A θ ). N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 21 / 36

Cocycles and actions on K(H) An ω-representation g u g of G on a Hilbert space H gives an ordinary action α ω : G Aut(K(H)), by the formula α ω g (a) = u g au g. (When checking αg ω αh ω = αω gh, the extra factors ω(g, h) in u g u h = ω(g, h)u gh cancel out.) Moreover, we have C (G, K(H), α ω ) = C (G, ω) K(H) (using the complex conjugate cocycle). In the reverse direction, if α: G Aut(K(H)) is an action, then choosing unitaries u g such that α g = Ad(u g ) gives an ω-representation for some ω. Moreover, cohomology classes of cocycles correspond to exterior equivalence classes of actions. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 22 / 36

Homotopy There are suitable notions of homotopy of cocycles and of actions on K(H), and homotopic cocycles correspond to homotopic actions. We have therefore reduced our problem to showing that two homotopic actions of G on K(H) give crossed products with the same K-theory. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 23 / 36

Reduction to compact subgroups Roughly speaking, the Baum-Connes conjecture with coefficients reduces the computation of K (C (G, A, α)) to the computation of K (C (L, A, α L )) for compact subgroups L of G. We use the following version of this statement, which follows from work of Chabert, Echterhoff, and Oyono-Oyono. The group K top (G; A) is as in the Baum-Connes Conjecture with coefficients. Theorem Let G be a second countable locally compact group and let A and B be C*-algebras with actions of G. Let z KK0 G (A, B) have the property that for all compact subgroups L of G, the Kasparov product with the restriction res G L (z) KK 0 L (A, B) induces bijective homomorphisms KK L (C, A) KK L (C, B). Then the Kasparov product with z induces an isomorphism K top (G; A) = K top (G; B). N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 24 / 36

Reduction to compact subgroups (continued) We take z in the previous theorem to be evaluation of a homotopy defined over [0, 1] at a point of [0, 1]. The group G is taken to be Z 2 Z k, which is amenable, so satisfies the Baum-Connes conjecture with coefficients by Higson-Kasparov. We have therefore reduced our problem to showing that two homotopic actions of a finite group L on K(H) give crossed products with the same K-theory. This is not true for homotopic actions of a finite group on a general C*-algebra, even a general commutative unital C*-algebra. However, for homotopies of actions on K(H), we translate back to S 1 -valued 2-cocycles, and we find that the cohomology group H 2 (L, S 1 ) is discrete. Thus (omitting significant work), all homotopies are trivial. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 25 / 36

K (C (Z k, A θ )) is independent of θ Conclusion: K (C (Z k, A θ )) is independent of θ. So do the computation at θ = 0. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 26 / 36

Original argument There is a continuous field of C*-algebras over S 1 whose fiber over exp(2πiθ) is A θ. Its section algebra is the C*-algebra of the discrete Heisenberg group H, the group with generators u, v, and z such that z is central and vu = zuv. The crossed products C (Z k, A θ ) are the fibers of a continuous field over (the double cover of) S 1 whose section algebra is C (H Z k ) for a suitable action of Z k on H. The original argument was to compute K (C (H Z k )) explicitly (using the Baum-Connes Conjecture, since the group is amenable). A lot of work was then required to get back to K (C (Z k, A θ )). N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 27 / 36

UCT For classification, we also need to know that C (Z k, A θ ) satisfies the Universal Coefficient Theorem. This is also proved using the Baum-Connes Conjecture. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 28 / 36

Tracial rank zero We now outline the second step: classifiability. Recall: Theorem (Bratteli) Let A be a separable C*-algebra. Then A is AF if and only if for every finite set S A and every ε > 0, there exists a finite dimensional subalgebra F A such that dist(a, F ) < ε for all a S. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 29 / 36

Definition (Lin) Let A be a simple separable unital C*-algebra. Then A has tracial rank zero (is tracially AF) if for every finite set S A, every ε > 0, and every nonzero positive element x A, there is a projection p A and a finite dimensional unital subalgebra F pap (that is, p is the identity of F ) such that: 1 dist(pap, F ) < ε for all a S. 2 pa ap < ε for all a S. 3 1 p is Murray-von Neumann equivalent to a projection in xax. If we can always take p = 1, this says that A is AF. If A has enough tracial states, we can replace the last condition by τ(1 p) < ε for all tracial states τ. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 30 / 36

Lin s classification theorem Theorem (Lin) Let A and B be nuclear simple separable unital C*-algebras with tracial rank zero and satisfying the Universal Coefficient Theorem. Suppose (K 0 (A), K 0 (A) +, [1 A ], K 1 (A)) = K 0 (B), K 0 (B) +, [1 B ], K 1 (B)). (That is, A and B have the same Elliott invariant.) Then A = B. Let θ R \ Q. It is a consequence of the Elliott-Evans Theorem (A θ is a simple AT algebra with real rank zero) and work of Lin that A θ has tracial rank zero. We have to somehow get from this fact to the statement that C (Z k, A θ ) has tracial rank zero. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 31 / 36

The tracial Rokhlin property Definition Let A be an infinite dimensional finite simple separable unital C*-algebra, and let α: G Aut(A) be an action of a finite group G on A. We say that α has the tracial Rokhlin property if for every finite set F A, every ε > 0, and every positive element x A with x = 1, there are mutually orthogonal projections e g A for g G such that: 1 α g (e h ) e gh < ε for all g, h G. 2 e g a ae g < ε for all g G and all a F. 3 With e = g G e g, the projection 1 e is Murray-von Neumann equivalent to a projection in xax. (If A is not finite, one must add an extra condition.) If we can always take p = 1, this becomes the Rokhlin property, which has a long history. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 32 / 36

Crossed products It is not true (Blackadar) that crossed products of AF algebras by finite groups are AF. However, the following theorem is not hard to prove: Theorem (from earlier work) Let A be a unital AF algebra. Let α: G Aut(A) be an action of a finite group G on A which has the Rokhlin property. Then C (G, A, α) is an AF algebra. If one takes care of the leftover projections carefully (a result of Jeong-Osaka is needed here), one can use similar methods to get: Theorem (from earlier work) Let A be an infinite dimensional simple separable unital C*-algebra with tracial rank zero. Let α: G Aut(A) be an action of a finite group G on A which has the tracial Rokhlin property. Then C (G, A, α) has tracial rank zero. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 33 / 36

Actions with the tracial Rokhlin property No action of any nontrivial finite group on any irrational rotation algebra has the Rokhlin property. However: Theorem Let A be a simple separable unital C*-algebra with tracial rank zero, and suppose that A has a unique tracial state τ. Let G be a finite group, and let α: G Aut(A) be an action of G on A. Let π τ : A B(H τ ) be the Gelfand-Naimark-Segal representation associated with τ. Then α has the tracial Rokhlin property if and only if α g is an outer automorphism of π τ (A) for every g G \ {1}. It is not difficult to verify that the hypotheses are satisfied for our actions of Z k on A θ. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 34 / 36

Conclusion Let θ R \ Q. Since the action of Z k on A θ has the tracial Rokhlin property and since A θ has tracial rank zero, we conclude that C (Z k, A θ ) has tracial rank zero. We noted above that C (Z k, A θ ) satisfies the Universal Coefficient Theorem. Therefore Lin s classification theorem applies. The K-theory computations showed that K 0 (C (Z k, A θ )) is torsion free and K 1 (C (Z k, A θ )) = 0. Combining this with tracial rank zero, one shows that there is a simple unital AF algebra B k,θ with the same Elliott invariant. So Lin s classification theorem implies that C (Z k, A θ ) = B k,θ, and in particular is AF. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 35 / 36

Two open problems Problem Give a direct proof that C (Z k, A θ ) is AF. Problem Is the action of Z k on A θ a direct limit action for some representation of A θ as an AT algebra? For k = 2, Walters has given a positive solution to the second problem. This also gives a solution to the first problem, but the case k = 2 of the first problem was already known. N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 36 / 36