Let F ⊆ SL 2 (Z) be a finite subgroup (necessarily isomorphic to one of Z 2 , Z 3 , Z 4 , or Z 6 ), and let F act on the irrational rotational algebra A θ via the restriction of the canonical action of SL 2 (Z). Then the crossed product A θ ⋊α F and the fixed point algebra A F θ are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of Z 2 on any simple d-dimensional noncommutative torus A Θ . Along the way, we prove a number of general results which should have useful applications in other situations.1 2 ECHTERHOFF, LÜCK, PHILLIPS, AND WALTERS Theorem 0.1 (Theorems 4.9, 6.3, and 6.4). Let F be any of the finite subgroups Z 2 , Z 3 , Z 4 , Z 6 ⊆ SL 2 (Z) with generators given as above and let θ ∈ R Q. Then the crossed product A θ ⋊ α F is an AF algebra. For all θ ∈ R we haveFor F = Z k for k = 2, 3, 4, 6, the image of K 0 (A θ ⋊ α F ) under the canonical tracial state on A θ ⋊ α F (which is unique) is equal to 1 k (Z+θZ). As a consequence, A θ ⋊ α Z k is isomorphic to A θ ′ ⋊ α Z l if and only if k = l and θ ′ = ±θ mod Z.

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.

No abstract

For d = 2, 3, . . . and p ∈ [1, ∞), we define a class of representations ρ of the Leavitt algebra L d on spaces of the form L p (X, µ), which we call the spatial representations. We prove that for fixed d and p, the Banach algebrais the same for all spatial representations ρ. When p = 2, we recover the usual Cuntz algebra O d . We give a number of equivalent conditions for a representation to be spatial. We show that for distinct p 1 , p 2 ∈ [1, ∞) and arbitrary d 1 , d 2 ∈ {2, 3, . . .}, there is no nonzero continuous homomorphism from O p 1 d 1 to O p 2 d 2 .

Abstract. We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form C(X, Mn) many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies in particular that if A is a separable C*-algebra whose irreducible representations all have dimension at most N < ∞, and if for each n the space of n-dimensional irreducible representations has finite covering dimension, then A is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it.Consequences for simple direct limits of recursive subhomogeneous algebras, with applications to the transformation group C*-algebras of minimal homeomorphisms, will be given in a separate paper.