The Tracial Class Property for Crossed Products by Finite Group Actions

and Applied Analysis 3 Lemma 2.2 see 5, Theorem 3.3 , 6, Theorem 3.3 . Let A be a simple unital C∗-algebra. If TRR A 0, then RR A 0. If Tsr A 1 and has the (SP)-property, then tsr A 1. Definition 2.3 see 13, Definition 1.2 . 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 ε > 0, every finite set F ⊆ A, every positive element b ∈ A, there are mutually orthogonal projections {eg : 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 , 1 − e ≤ b . Lemma 2.4 see 13, Corollary 1.6 . LetA 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 which has the tracial Rokhlin property. Then A ×α G is simple. Lemma 2.5 see 20, Theorem 4.2 . Let A be a simple unital C∗-algebra with the (SP)-property, and let α : G → Aut A be an action of a discrete group G on A. Suppose that the normal subgroup N {g ∈ G | αg is inner on A} of G is finite; then any nonzero hereditary C∗-subalgebra of the crossed product A ×α G has a nonzero projection which is Murray-von Neumann equivalent to a projection in A ×α N. If the action α : G → Aut A has the tracial Rokhlin property, then each αg is outer for all g ∈ G \ {1}. So N {g ∈ G | αg is inner on A} {1}. Since A ×α N A ×α {1} ∼ A, by Lemma 2.5 we have the following lemma. Lemma 2.6. LetA be an infinite dimensional finite simple separable unital C∗-algebra with the (SP)property, and let α : G → Aut A be an action of a finite group G on A which has the tracial Rokhlin property; then any nonzero hereditary C∗-subalgebra of the crossed product A ×α G has a nonzero projection which is Murray-von Neumann equivalent to a projection in A. Lemma 2.7 see 19, Lemma 3.5.6 . Let A be a simple C∗-algebra with the (SP)-property, and let p, q ∈ A be two nonzero projections. Then there are nonzero projections p1 ≤ p, q1 ≤ q such that p1 q1 . Definition 2.8 see 14, Definition 1.1 . Let C be a class of separable unital C∗-algebras. We say that C is finitely saturated if the following closure conditions hold. 1 If A ∈ C and B ∼ A, then B ∈ C. 2 If Ai ∈ C for i 1, 2, . . . , n, then ⊕n k 1Ak ∈ C. 3 If A ∈ C and n ∈ N, thenMn A ∈ C. 4 If A ∈ C and p ∈ A is a nonzero projection, then pAp ∈ C. Moreover, the finite saturation of a class C is the smallest finitely saturated class which contains C. Definition 2.9 see 14, Definition 1.2 . Let C be a class of separable unital C∗-algebras. We say that C is flexible if. 4 Abstract and Applied Analysis 1 for every A ∈ C, every n ∈ N, and every nonzero projection p ∈ Mn A , the corner pMn A p is semiprojective and finitely generated; 2 for every A ∈ C and every ideal I ⊆ A, there is an increasing sequence I1 ⊆ I2 ⊆ · · · of ideals of A such that ∪n 1In I and such that for every n the C∗-algebra A/In is in the finite saturation of C. Example 2.10. 1 Let C {⊕ni 1Mk i | n, k i ∈ N}; that is, C contains all finite dimensional algebras. C is finitely saturated and flexible. 2 Let C {⊕ni 1C Xi,Mk i | n, k i ∈ N; each Xi is a closed subset of the circle}. We can show that C is finitely saturated and flexible. 3 Let C {f ∈ ⊕ni 1C 0, 1 ,Mk i | n, k i ∈ N, f 0 is scalar}. We can also show that C is finitely saturated and flexible. 4 For some d ∈ N, let Cd contain all the C∗-algebras ⊕n i 1piC Xi,Mk i pi, where n, k i ∈ N, each pi is a nonzero projection in C Xi,Mk i , and each Xi is a compact metric space with covering dimension at most d. The class Cd is not flexible for d / 0 see 14 Example 2.9 . Definition 2.11 see 16, Definition 1.4 . Let C be a class of separable unital C∗-algebras. A unital approximate C-algebra is a C∗-algebra which is isomorphic to an inductive limit limn→∞ An, φn , where each An is in the finite saturation of C and each homomorphism φn : An → An 1 is unital. Definition 2.12 see 14, Definition 1.5 . Let C be a class of separable unital C∗-algebras. Let A be a separable unital C∗-algebra. We say that A is a unital local C-algebra if, for every ε > 0 and every finite subset F ⊂ A, there is a C∗-algebra B in the finite saturation of C and a ∗-homomorphism φ : B → A such that a ∈ε φ B for all a ∈ F. By 14 Proposition 1.6, if C is a finitely saturated flexible class of separable unital C∗algebras, then every unital local C-algebra is a unital approximate C-algebra. The converse is clear. LetC be a class as 1 of Example 2.10. Then a unital AF-algebra is a unital approximate C-algebra and is a unital local C-algebra. LetC be a class as 2 of Example 2.10. Then a unital AT-algebra is a unital approximate C-algebra and is a unital local C-algebra. Definition 2.13. Let A be a simple unital C∗-algebra, and let C be a class of separable unital C∗-algebra. We say that A is a tracial C-algebra if, for any ε > 0, any finite subset F ⊂ A, and any nonzero positive element b ∈ A, there exist a nonzero projection p ∈ A, a C∗-algebra B in the finite saturation of C, and a ∗-homomorphism φ : B → A with 1φ B p, such that 1 ‖pa − ap‖ < ε for any a ∈ F, 2 pap ∈ε φ B for all a ∈ F, 3 1 − p ≤ b in A. Using the similar proof of Lemma 3.6.5 of 19 about the tracial rank of unital hereditary C∗-subalgebras of a simple unital C∗-algebra, we get the following one. Lemma 2.14. LetC be any finitely saturated class of separable unitalC∗-algebras. Let p be a projection in a simple unital C∗-algebra A with the (SP)-property. If A is a tracial C-algebra, so also is pAp. Abstract and Applied Analysis 5and Applied Analysis 5 For n ∈ N, δ > 0, a unital C∗-algebra A, if wi,j , for 1 ≤ i, j ≤ n, are elements of A, such that ‖wi,j‖ ≤ 1 for 1 ≤ i, j ≤ n, such that ‖w∗ i,j − wj,i‖ < δ for 1 ≤ i, j ≤ n, such that ‖wi1,j1wi2,j2 − δj1,i2wi1,j2‖ < δ for 1 ≤ i1, i2, j1, j2 ≤ n, and such that wi,i are mutually orthogonal projections, we say that wi,j 1 ≤ i, j ≤ n form a δ-approximate system of n × n matrix units in A. By perturbation of projections see Theorem 2.5.9 of 19 , we have Lemma 2.15. Lemma 2.15. For any n ∈ N, any ε > 0, there exists δ δ n, ε > 0 such that, whenever fi,j 1≤i,j≤n is a system of matrix units for Mn, whenever B is a unital C∗-algebra, and whenever wi,j , for 1 ≤ i, j ≤ n, are elements of B which form a δ-approximate system of n× n matrix units, then there exists a ∗-homomorphism φ : Mn → B such that φ fi,i wi,i for 1 ≤ i ≤ n and ‖φ fi,j − wi,j‖ < ε for 1 ≤ i, j ≤ n.