The Tracial Rokhlin Property for Actions of Finite Groups on C*-algebras

We define “tracial” analogs of the Rokhlin property for actions of finite groups, approximate representability of actions of finite abelian groups, and of approximate innerness. We prove the following four analogs of related “nontracial” results. • The crossed product of an infinite dimensional simple separable unital C*-algebra with tracial rank zero by an action of a finite group with the tracial Rokhlin property again has tracial rank zero. • An outer action of a finite abelian group on an infinite dimensional simple separable unital C*algebra has the tracial Rokhlin property if and only if its dual is tracially approximately representable, and is tracially approximately representable if and only if its dual has the tracial Rokhlin property. • If a strongly tracially approximately inner action of a finite cyclic group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property, then it is tracially approximately representable. • An automorphism of an infinite dimensional simple separable unital C*-algebra A with tracial rank zero is tracially approximately inner if and only if it is the identity on K0(A) mod infinitesimals. 0. Introduction. Tracially AF C*-algebras, now known as C*-algebras with tracial rank zero, were introduced in [14]. Roughly speaking, a C*-algebra has tracial rank zero if the local approximation characterization of AF algebras holds after cutting out a “small” approximately central projection. The term “tracial” comes from the fact that, in good cases, a projection p is “small” if τ (p) < ε for every tracial state τ on A. The classification [17] of simple separable nuclear C*-algebras with tracial rank zero and satisfying the Universal Coefficient Theorem can be regarded as a vast generalization of the classification of AF algebras. This success suggests that one consider “tracial” versions of other C*-algebra concepts. In this paper, motivated by applications to particular crossed products (see [23] and [7]), we formulate and prove “tracial” versions of the following theorems: • The crossed product of an AF algebra by an action of a finite group with the Rokhlin property is again AF (Theorem 2.2). • An action of a finite abelian group on a unital C*-algebra has the Rokhlin property if and only if its dual is approximately representable, and is approximately representable if and only if its dual has the Rokhlin property. (Lemma 3.8 of [10]). Manuscript received September 30, 2008. Research supported in part by NSF grants DMS 0070776 and DMS 0302401. American Journal of Mathematics 132 (2010), 0–00. c © 2010 by The Johns Hopkins University Press. 1 2 N. CHRISTOPHER PHILLIPS • If an approximately inner action of a finite cyclic group on a unital C*algebra has the Rokhlin property, then it is approximately representable (Proposition 4.1). • An automorphism of an AF algebra A is approximately inner if and only if it is the identity on K0(A) (part of Theorem 3.1 of [2]). Our results are: • The crossed product of an infinite dimensional simple separable unital C*algebra with tracial rank zero by an action of a finite group with the tracial Rokhlin property again has tracial rank zero (Theorem 2.6). • An outer action of a finite abelian group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property if and only if its dual is tracially approximately representable, and is tracially approximately representable if and only if its dual has the tracial Rokhlin property (Theorem 3.11). • If a strongly tracially approximately inner action of a finite cyclic group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property, then it is tracially approximately representable (Theorem 4.6). • An automorphism of an infinite dimensional simple separable unital C*algebra A with tracial rank zero is tracially approximately inner if and only if it is the identity on K0(A) mod infinitesimals (Theorem 6.4). The first three of these results were chosen because they are used in the proof [23] that every simple higher dimensional noncommutative torus is an AT algebra. (We have not found the “nontracial” versions of the first and third results in the literature. Therefore they are also proved in this paper.) The last is related to our effort to find the “right” definition of a tracially approximately inner automorphism. Annoyingly, the strongly tracially approximately inner automorphisms (as in the third result) probably don’t form a group. On the other hand, an automorphism of an infinite dimensional simple separable unital C*-algebra with tracial rank zero which is tracially approximately inner and has finite order must in fact be strongly tracially approximately inner. (Combine Proposition 6.2 and Theorem 6.6.) In retrospect, the following motivation is perhaps better. In [10] and [11], Izumi has started an intensive study of finite group actions with the Rokhlin property, which, to minimize confusion, we call here the strict Rokhlin property. The strict Rokhlin property imposes severe restrictions on the relation between the K-theory of the original algebra, the action of the group on this K-theory, and the K-theory of the crossed product. See especially Section 3 of [11]. Since actions with the strict Rokhlin property are so rare, a less restrictive version of the Rokhlin property is needed. We give some examples; in them, we write Zn for Z/nZ. • The flip action of Z2 on A⊗A, for an infinite dimensional simple separable unital C*-algebra A, often has the tracial Rokhlin property, but probably almost never has the strict Rokhlin property. See [21]. • Let A be a simple higher dimensional noncommutative torus, with standard unitary generators u1, u2, . . . , ud. Consider the automorphism which sends uk to THE TRACIAL ROKHLIN PROPERTY 3 exp (2πi/n)uk, and fixes uj for j = k. This automorphism generates an action of Zn which has the tracial Rokhlin property, but for n > 1 never has strict Rokhlin property. The fact that this action has the tracial Rokhlin property plays a key role in the classification [23] of simple higher dimensional noncommutative toruses. • Again let A be a simple higher dimensional noncommutative torus, with standard unitary generators u1, u2, . . . , ud. The flip automorphism uk → uk generates an action of Z2 which has the tracial Rokhlin property, but never has the strict Rokhlin property. See [7], where this fact is used to prove that the crossed product by the flip action is always AF. • The standard actions of Z3, Z4, and Z6 on an irrational rotation algebra all have the tracial Rokhlin property, but never have the strict Rokhlin property. In [7], this is used to prove that the crossed products are always AF algebras. Of course, one can’t expect classification results for actions of the kind found in [10] and [11]. This paper is devoted to the general theory. In [24], we give several useful criteria for the tracial Rokhlin property, and we give a number of examples of actions of Z2 on C*-algebras with tracial rank zero (mostly AF algebras) which do and do not have the tracial Rokhlin property, and are or are not tracially approximately representable. Further examples, and results for C*-algebras with finite but nonzero tracial rank, will appear in [21]. The main applications, already mentioned above, are in [23] and [7]. We also point out that the tracial Rokhlin property has a generalization to integer actions, considered in [19] and with applications given in [20]. Presumably there is a useful generalization to other countable amenable groups. This paper replaces Sections 1 through 4 and Section 11 of the unpublished long preprint [22]. The material of Sections 5 through 7 there will appear in [23], a greatly improved version of Sections 8 through 10 will appear in [7], and an improved and expanded version of the material in Sections 12 and 13 will appear in [24]. We give the theory for actions of finite groups, or of finite abelian groups, as appropriate; in [22], only finite cyclic groups were considered. The definition of the tracial Rokhlin property given here differs slightly from that in [22]; see Remark 1.3 and the following discussion for details. There is no difference for actions on simple unital C*-algebras with tracial rank zero. The definition of tracial approximate innerness (Definition 5.1) has been greatly improved; more automorphisms satisfy the condition than satisfied the condition in [22], and with the new definition the tracially approximately inner automorphisms form a group. (See Theorem 5.8.) A variant of the original definition appears in Definition 4.2 here, where the condition is called strong tracial approximate innerness. The material in Section 3 on tracial approximate representability is mostly new, although it was motivated by one of the key results in [22] and its parallel with Lemma 3.8 of [10]. This paper is organized as follows. In Section 1 we introduce the tracial Rokhlin property and prove some basic properties. Some of the lemmas will be used repeatedly in connection with the tracial versions of other properties. 4 N. CHRISTOPHER PHILLIPS Section 2 contains the proofs that the crossed product of an AF algebra by an action of a finite group with the Rokhlin property is again AF, and that the crossed product of an infinite dimensional simple separable unital C*-algebra with tracial rank zero by an action of a finite group with the tracial Rokhlin property again has tracial rank zero. In Section 3 we treat tracial approximate representability for actions of finite abelian groups, and prove the duality between this property and the tracial Rokhlin property, corresponding to Lemma 3.8 of [10]. In Section 4 we introduce strongly tracially approximately inner automorphism, and prove that if a strongly tracially approximately inner action of a finite cyclic group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property, then it is tracially approximately representable. We also prove the “nontracial” analog of this result. Sections 5 and 6 treat tracially approximately inner automorphisms. We show that they form a group. We prove that they act trivially on the tracial state space and on K0 mod infinitesimals, and give a converse when the algebra is simple with tracial rank zero. We also prove that if the the algebra is simple with tracial rank zero, then a tracially approximately inner automorphism of finite order is necessarily strongly tracially approximately inner. We use the following notation. We write p q to mean that the projection p is Murray-von Neumann equivalent to a subprojection of q, and p ∼ q to mean that p is Murray-von Neumann equivalent to q. Also, [a, b] denotes the additive commutator ab− ba. If A is a C*-algebra and α: G → Aut (A) is a group action, we write Aα for the fixed point algebra. In most arguments dealing with an arbitrary finite subset F of a C*-algebra, we will normalize and assume that all elements of F have norm at most 1. Acknowledgments. We are grateful to Hanfeng Li for valuable comments, and to Hiroyuki Osaka for a careful reading and suggesting improvements to several of the proofs. We are also grateful to Dawn Ashley for catching a number of misprints and minor mistakes. 1. The tracial Rokhlin property. In this section we introduce the tracial Rokhlin property. We observe several elementary relations and consequences, and we prove several useful equivalent formulations. Some of the technical lemmas will be repeatedly used in connection with other “tracial” properties. We begin with Izumi’s definition of the Rokhlin property. To emphasize the difference, we call it the strict Rokhlin property here. Definition 1.1. Let A be a 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 strict Rokhlin property if for every finite set F ⊂ A and every ε > 0, there are mutually orthogonal projections eg ∈ A for g ∈ G such that: (1) ‖αg(eh) − egh‖ < ε for all g, h ∈ G. (2) ‖ega − aeg‖ < ε for all g ∈ G and all a ∈ F. (3) ∑ g∈G eg = 1. THE TRACIAL ROKHLIN PROPERTY 5 Izumi’s definition (Definition 3.1 of [10]) is actually in terms of central sequences. Thus, it yields not mutually orthogonal projections but elements bg ∈ A such that ‖bgbh−δg,hbg‖ < ε for g, h ∈ G, such that ‖bg−bg‖ < ε for g ∈ G, and such that ∥∥∥1 −∑g∈G bg∥∥∥ < ε. However, with n = card (G), using semiprojectivity of Cn (see Lemma 14.1.5, Theorem 14.2.1, Theorem 14.1.4, and Definition 14.1.1 of [18]) and a suitably smaller choice of ε, one easily sees that the definition above is equivalent to Definition 3.1 of [10]. If α is approximately inner, requiring ∑ g∈G eg = 1 forces [1A] ∈ K0(A) to be divisible by the order of G, and therefore rules out many C*-algebras of interest. In fact, the strict Rokhlin property imposes much more stringent conditions on the K-theory. Theorem 3.3 and Lemma 3.2(1) of [11] show that if a nontrivial finite group G acts on a simple unital C*-algebra A in such a way that the induced action on K∗(A) is trivial, and if one of K0(A) and K1(A) is a nonzero free abelian group, then α does not have the strict Rokhlin property. Theorem 3.3 and the discussion preceding Theorem 3.4 of [11] show that if in addition G is cyclic of order n, then the strict Rokhlin property implies that K∗(A) is uniquely n-divisible. It follows that the actions considered in our main applications (to simple higher dimensional noncommutative toruses [23] and irrational rotation algebras [7]) never have the strict Rokhlin property. We now give the definition of the tracial Rokhlin property. The difference is that we do not require that ∑ g∈G eg = 1, only that 1 − ∑ g∈G eg be “small” in a tracial sense. Of course, ∑ g∈G eg = 1 is allowed, in which case Conditions (1.2) and (1.2) in the definition are vacuous. Definition 1.2. Let A be an infinite dimensional 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 eg ∈ A for g ∈ G such that: (1) ‖αg(eh) − egh‖ < ε for all g, h ∈ G. (2) ‖ega − aeg‖ < ε for all g ∈ G and all a ∈ F. (3) With e = ∑ g∈G eg, the projection 1−e is Murray-von Neumann equivalent to a projection in the hereditary subalgebra of A generated by x. (4) With e as in (1.2), we have ‖exe‖ > 1 − ε. Remark 1.3. Our original definition, in [22], in addition specified a positive integer N, and required the following condition instead of (1.2): (4′) For every g ∈ G, there are N mutually orthogonal projections f1, f2, . . . , fN ≤ eg, each of which is Murray-von Neumann equivalent to the projection 1−e of (1.2). As we will see in Lemma 1.16 below, when A is finite, Condition (1.2) in Definition 1.2 is unnecessary. When A has finite tracial topological rank in the sense of [15], it is not hard to see that Definition 1.2 implies the definition in 6 N. CHRISTOPHER PHILLIPS Remark 1.3. In general, the situation is less clear, and it might be necessary to use both Condition (1.2) and Condition (4′), especially for nonsimple C*-algebras. We postpone further discussion to Section 4 of [24]. Remark 1.4. If an action of a finite group G on an infinite dimensional simple separable unital C*-algebra A has the strict Rokhlin property, then it has the tracial Rokhlin property. LEMMA 1.5. Let A be an infinite dimensional simple separable unital C*algebra, and let α: G → Aut (A) be an action of a finite group G on A which has the tracial Rokhlin property. Then αg is outer for every g ∈ G \ {1}. Proof. Let g ∈ G \ {1} and let u ∈ A be unitary. We prove that αg = Ad (u). Apply Definition 1.2 with F = {u}, with ε = 2 , and with x = 1. Then e1 and eg are orthogonal nonzero projections, so ‖αg(e1) − ue1u‖ ≥ ‖eg − e1‖ − ‖αg(e1) − eg‖ − ‖ue1u − e1‖ > 0. Therefore αg = Ad (u). COROLLARY 1.6. Let A be an infinite dimensional simple separable unital C*algebra, and 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) is simple. Proof. In view of Lemma 1.5, this follows from Theorem 3.1 of [13]. For the tracial Rokhlin property to be likely to hold, the C*-algebra must have a reasonable number of projections. For reference, we recall here the definition of the property that seems most relevant. Definition 1.7. Let A be a C*-algebra. We say that A has Property (SP) if every nonzero hereditary subalgebra in A contains a nonzero projection. We state here some results about simple C*-algebras with Property (SP) that will be used repeatedly in this paper. LEMMA 1.8. Let A be a C*-algebra, and let c ∈ A. Then for any projection p ∈ cAc∗, there exists a projection q ∈ c∗Ac such that p ∼ q. Proof. This is essentially in Section 1 of [5]. The details can be found in the proof of Lemma 4.1 of [19]. The following lemma is essentially Lemma 3.1 of [14], but no proof is given there. LEMMA 1.9. Let A be a simple C*-algebra with Property (SP). Let B ⊂ A be a nonzero hereditary subalgebra, and let p ∈ A be a nonzero projection. Then there is a nonzero projection q ∈ B such that q p. THE TRACIAL ROKHLIN PROPERTY 7 Proof. Choose a nonzero positive element a ∈ B. Since A is simple, there exists x ∈ A such that c = axp is nonzero. Choose a nonzero projection q ∈ cAc∗. Then q ∈ B, and Lemma 4.1 of [19] (or Lemma 1.8) provides a projection e ≤ p such that q ∼ e. LEMMA 1.10. Let A be an infinite dimensional simple unital C*-algebra with Property (SP). Let B ⊂ A be a nonzero hereditary subalgebra, and let n ∈ N. Then there exist nonzero Murray-von Neumann equivalent mutually orthogonal projections p1, p2, . . . , pn ∈ B. Proof. Since A is unital and infinite dimensional, it is not isomorphic to the algebra of compact operators on any Hilbert space. The lemma is then immediate from Lemma 3.2 of [14]. (The result from [1] is on page 61 of that reference. See above for the proof of Lemma 3.1 of [14].) LEMMA 1.11. Let A be an infinite dimensional simple unital C*-algebra, and let n ∈ N. Then A has Property (SP) if and only if Mn⊗A has Property (SP). Moreover, in this case, for every nonzero hereditary subalgebra B ⊂ Mn ⊗ A, there exists a nonzero projection p ∈ A such that 1 ⊗ p is Murray-von Neumann equivalent to a projection in B. Proof. Since A is isomorphic to a hereditary subalgebra in Mn ⊗ A, it is obvious that if Mn ⊗ A has Property (SP), then so does A. For the converse and the last statement, choose a nonzero element x ∈ B. Let (ej,k)1≤j,k≤n be a system of matrix units for Mn. Choose j such that (ej,j ⊗1)x = 0. Then C = (ej,j ⊗ 1)x(Mn ⊗ A)x∗(ej,j ⊗ 1) is a nonzero hereditary subalgebra in (ej,j⊗1)(Mn⊗A)(ej,j⊗1) ∼= A. Because A has Property (SP), there exists a nonzero projection f ∈ A such that ej,j ⊗ f ∈ C. By Lemma 1.10, there exist nonzero Murray-von Neumann equivalent mutually orthogonal projections f1, f2, . . . , fn ∈ A with fj ≤ f for all k. Then