Abstract. We introduce a general class of automorphisms of rotation algebras, the noncommutative Furstenberg transformations. We prove that fully irrational noncommutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by these automorphisms have stable rank one, real rank zero, and order on projections determined by traces (Blackadar’s Second Fundamental Comparability Question). We also prove that several classes of simple quotients of the C*-algebras of discrete subgroups of five dimensional nilpotent Lie groups, considered by Milnes and Walters, are crossed products of simple C*-algebras (C*-algebras of minimal ordinary Furstenberg transformations) by automorphisms which have the tracial Rokhlin property. It follows that these algebras also have stable rank one, real rank zero, and order on projections determined by traces. Furstenberg introduced in [7] a family of homeomorphisms of S 1 ×S 1, now called

Abstract. We show that, if A is a separable simple unital C ∗-algebra which absorbs the Jiang–Su algebraZ tensorially and which has real rank zero and finite decomposition rank, then A is tracially AF in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of C ∗-algebras as above which, additionally, satisfy the Universal Coefficients Theorem. In particular, such algebras are completely determined by their ordered K-theory. They are approximately homogeneous of topological dimension less than or equal to 3, approximately subhomogeneous of topological dimension at most 2 and their decomposition rank also is no greater than 2. It is the aim of the Elliott classification program to find complete K-theoretic invariants for separable simple nuclear C ∗-algebras. For purely infinite C ∗-algebras, this task was accomplished by Kirchberg and Phillips, and there are numerous classification results for special inductive limits of (sub)homogeneous C ∗-algebras,

Abstract. We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Let A be a stably finite simple unital C*-algebra, and let α be an automorphism of A which has the tracial Rokhlin property. Suppose A has real rank zero and stable rank one, and suppose that the order on projections over A is determined by traces. Then the crossed product algebra C ∗ (Z, A, α) also has these three properties. We also present examples of C*-algebras A with automorphisms α which satisfy the above assumptions, but such that C ∗ (Z, A, α) does not have tracial rank zero. We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally

Abstract. We show that, if a simple C ∗-algebra A is topologically finitedimensional in a suitable sense, then not only K0(A) has certain good properties, but A is even accessible to Elliott’s classification program. More precisely, we prove the following results: If A is simple, separable and unital with finite decomposition rank and real rank zero, then K0(A) is weakly unperforated. If A has finite decomposition rank, real rank zero and the space of extremal tracial states is compact and zero-dimensional, then A has stable rank one and tracial rank zero. As a consequence, if B is another such algebra, and if A and B have isomorphic Elliott invariants and satisfy the Universal coefficient theorem, then they are isomorphic. In the case where A has finite decomposition rank and the space of extremal tracial states is compact and zero-dimensional, we also give a criterion (in terms of the ordered K0-group) for A to have real rank zero. As a byproduct, we show that there are examples of simple, stably finite and quasidiagonal

Abstract. This is a detailed survey on the QWEP conjecture and Connes’ embedding problem. Most of contents are taken from Kirchberg’s paper [Invent. Math. 112 (1993)]. 1.

Abstract. We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C ∗-algebras. The concept is particularly relevant for Elliott’s program to classify nuclear C ∗-algebras by K-theory data. We study some of its properties and show that a simple unital C ∗-algebra, which has locally finite decomposition rank, real rank zero and which absorbs the Jiang–Su algebra Z tensorially, has tracial rank zero in the sense of Lin. As a consequence, any such C ∗-algebra, if it additionally satisfies the Universal Coefficients Theorem, is approximately homogeneous of topological dimension at most 3. Our result in particular confirms the Elliott conjecture for the class of simple unital Z-stable ASH algebras with real rank zero. Moreover, it implies that simple unitalZ-stable AH algebras with real rank zero not only have slow dimension growth in the ASH sense, but even in the AH sense. This note is concerned with the stably finite real rank zero case of Elliott’s program

Abstract. Say that a separable, unital C ∗-algebra D ≇ C is strongly selfabsorbing if there exists an isomorphism ϕ: D → D ⊗ D such that ϕ and idD ⊗1D are approximately unitarily equivalent ∗-homomorphisms. We study this class of algebras, which includes the Cuntz algebras O2, O∞, the UHF algebras of infinite type, the Jiang–Su algebra Z and tensor products of O∞ with UHF algebras of infinite type. Given a strongly self-absorbing C ∗-algebra D we characterise when a separable C ∗-algebra absorbs D tensorially (i.e., is D-stable), and prove closure properties for the class of separable D-stable C ∗-algebras. Finally, we compute the possible K-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C ∗-algebras. Elliott’s program to classify nuclear C*-algebras via K-theoretic invariants (see [7] for an introductory overview) has met with considerable success since his seminal classification of approximately finite-dimensional (AF) algebras via the scaled ordered

Abstract. We show the existence of noncommutative random variables with finite free entropy but which do not generate a free group factor. In particular, this gives an example of variables X1,...,Xn such that δ(X1,..., Xn) = n while W ∗ (X1,...,Xn) ≇ L(Fn). 1.

Abstract. The Jiang–Su algebra Z has come to prominence in the classification program for nuclear C ∗-algebras of late, due primarily to the fact that Elliott’s classification conjecture predicts that all simple, separable, and nuclear C ∗-algebras with unperforated K-theory will absorb Z tensorially (i.e., will be Z-stable). There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable andZ-stable C ∗-algebras. We prove that virtually all classes of nuclear C ∗-algebras for which the Elliott conjecture has been confirmed so far, consist of Z-stable C ∗-algebras. This result follows in large part from the following theorem, also proved herein: separable and approximately divisible C ∗-algebras areZ-stable. The Jiang–Su algebra Z is a simple, separable, unital and nuclear C ∗-algebra KK-equivalent to C ([12]). Since its discovery in 1995 there has been a steady accumulation of evidence linking Z to Elliott’s program to classify separable, nuclear C ∗-algebras via K-theoretic invariants: in [12], Jiang and Su prove that simple,

Abstract. Let X be a finite CW complex and let h1, h2: C(X) → A be two unital homomorphisms, where A is a unital C ∗-algebra. We study the problem when h1 and h2 are approximately homotopy. We present a K-theoretical necessary and sufficient condition for them to be approximately homotopy under the assumption that A is a unital separable simple C ∗-algebra of tracial rank zero, or A is a unital purely infinite simple C ∗-algebra. When they are approximately homotopy, we also give a bound for the length of the homotopy. These results are also extended to the case that h1 and h2 are approximately multiplicative contractive completely positive linear maps. Suppose that h: C(X) → A is a monomorphism and u ∈ A is a unitary (with [u] = {0} in K1(A)). We prove that, for any ǫ> 0, and any compact subset F ⊂ C(X), there exists δ> 0 and a finite subset G ⊂ C(X) satisfying the following: if ‖[h(f), u] ‖ < δ and Bott(h, u) = {0}, then there exists a continuous rectifiable path {ut: t ∈ [0, 1]} such that u0 = u, u1 = 1A and ‖[h(g), ut] ‖ < ǫ for all g ∈ F and t ∈ [0,1]. (e 0.1)