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In this paper we study Cuntz--Pimsner algebras associated to $\mathrm{C}^*$-correspondences over commutative $\mathrm{C}^*$-algebras from the point of view of the $\mathrm{C}^*$-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite-dimensional infinite compact metric space $X$ twisted by a vector bundle, the resulting Cuntz--Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these $\mathrm{C}^*$-algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz--Pimsner algebra has stable rank one. Otherwise, it is purely infinite. For a Cuntz--Pimsner algebra of a minimal homeomorphism of an infinite compact metric space $X$ twisted by a line bundle over $X$, we introduce orbit-breaking subalgebras. With no assumptions on the dimension of $X$, we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of $X$ is finite, they are furthermore $\mathcal{Z}$-stable and hence classified by the Elliott invariant.

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We construct Cartan subalgebras in all classifiable stably finite C*-algebras. Together with known constructions of Cartan subalgebras in all UCT Kirchberg algebras, this shows that every classifiable simple C*-algebra has a Cartan subalgebra.

We prove that every unital stably finite simple amenable $C^*$-algebra $A$
with finite nuclear dimension and with UCT such that every trace is
quasi-diagonal has the property that $A\otimes Q$ has generalized tracial rank
at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is
classifiable in the sense of Elliott. In particular, if $A$ is a unital
separable simple $C^*$-algebra with finite decomposition rank which satisfies
the UCT, then $A$ is classifiable.

We introduce the concept of finitely coloured equivalence for unital
*-homomorphisms between C*-algebras, for which unitary equivalence is the
1-coloured case. We use this notion to classify *-homomorphisms from separable,
unital, nuclear C*-algebras into ultrapowers of simple, unital, nuclear,
Z-stable C*-algebras with compact extremal trace space up to 2-coloured
equivalence by their behaviour on traces; this is based on a 1-coloured
classification theorem for certain order zero maps, also in terms of tracial
data.
As an application we calculate the nuclear dimension of non-AF, simple,
separable, unital, nuclear, Z-stable C*-algebras with compact extremal trace
space: it is 1. In the case that the extremal trace space also has finite
topological covering dimension, this confirms the remaining open implication of
the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry
and topology, we derive a "homotopy equivalence implies isomorphism" result for
large classes of C*-algebras with finite nuclear dimension.

We present a classification theorem for a class of unital simple separable
amenable ${\cal Z}$-stable $C^*$-algebras by the Elliott invariant. This class
of simple $C^*$-algebras exhausts all possible Elliott invariant for unital
stably finite simple separable amenable ${\cal Z}$-stable $C^*$-algebras.
Moreover, it contains all unital simple separable amenable $C^*$-alegbras which
satisfy the UCT and have finite rational tracial rank.

We introduce the notion of the crossed product A⋊ X ℤ of a C * -algebra A by a Hilbert C * -bimodule X. It is shown that given a C * -algebra B which carries a semi-saturated action of the circle group (in the sense that B is generated by the spectral subspaces B 0 and B 1 ), then B is isomorphic to the crossed product B 0 ⋊ B 1 ℤ. We then present our main result, in which we show that the crossed products A⋊ X ℤ and B⋊ Y ℤ are strongly Morita equivalent to each other, provided that A and B are strongly Morita equivalent under an imprimitivity bimodule M satisfying X⊗ A M≃M⊗ B Y as A-B Hilbert C * -bimodules. We also present a six-term exact sequence for K-groups of crossed products by Hilbert C * -bimodules.

Let $X$ be an infinite compact metrizable space, and let $\sigma: X\to X$ be
a minimal homeomorphism. Suppose that $(X, \sigma)$ has zero mean topological
dimension. The associated C*-algebra $A=\mathrm{C}(X)\rtimes_\sigma\mathbb Z$
is shown to absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A\cong
A\otimes\mathcal Z$. This implies that $A$ is classifiable when $(X, \sigma)$
is uniquely ergodic.

We study minimal homeomorphisms (all orbits are dense) of the tori Tn, n≤4. The linear part of a homeomorphism φ of Tn is the linear mapping L induced by φ on the first homology group of Tn. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of L if φ minimal. We show that if φ is minimal and n≤4, then L is quasi-unipontent, that is, all of the eigenvalues of L are roots of unity and conversely if LGL(n,) is quasi-unipotent and 1 is an eigenvalue of L, then there exists a C∞ minimal skew-product diffeomorphism φ of Tn whose linear part is precisely L. We do not know whether these results are true for n≥5. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation.

We develop the concept of Rokhlin dimension for integer and for finite group
actions on C*-algebras. Our notion generalizes the so-called Rokhlin property,
which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin
dimension is prevalent and appears in cases in which the Rokhlin property
cannot be expected: the property of having finite Rokhlin dimension is generic
for automorphisms of Z-stable C*-algebras, where Z denotes the Jiang-Su
algebra. Moreover, crossed products by automorphisms with finite Rokhlin
dimension preserve the property of having finite nuclear dimension, and under a
mild additional hypothesis also preserve Z-stability. Finally, we prove a
topological version of the classical Rokhlin lemma: automorphisms arising from
minimal homeomorphisms of finite dimensional compact metrizable spaces always
have finite Rokhlin dimension.

The notion of extension of a given C*-category by a C*-algebra A is introduced. In the commutative case A=C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging to the initial category. It is shown that the Doplicher–Roberts algebra (DR-algebra in the following) associated to an object in the extension of a strict tensor C*-category is a continuous field of DR-algebras coming from the initial one. In the case of the category of the hermitian vector bundles over Ω the general result implies that the DR-algebra of a vector bundle is a continuous field of Cuntz algebras. Some applications to Pimsner C*-algebras are given.

Let X be an infinite, compact, metrizable space of finite covering dimension and h a minimal homeomorphism of X. We prove that the crossed product of C(X) by h absorbs the Jiang-Su algebra tensorially and has finite nuclear dimension. As a consequence, these algebras are determined up to isomorphism by their graded ordered K-theory under the necessary condition that their projections separate traces. This result applies, in particular, to those crossed products arising from uniquely ergodic homeomorphisms.

We present a general approach to a modular frame theory in C*-algebras and Hilbert C*-modules. The investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and by other bounded modular operators with suitable ranges. We obtain frame representations and decomposition theorems, as well as similarity and equivalence results for frames. Hilbert space frames and quasi-bases for conditional expectations of finite index on C*-algebras appear as special cases. Using a canonical categorical equivalence of Hilbert C*-modules over commutative C*-algebras and (F)Hilbert bundles the results find a reintepretation for frames in vector and (F)Hilbert bundles. Fields of applications are investigations on Cuntz-Krieger-Pimsner algebras, on conditional expectations of finite index, on various ranks of C*-algebras, on classical frame theory of Hilbert spaces (wavelet and Gabor frames), and others. 2001: In the introduction we refer to related publications in detail.

This book is directed towards graduate students that wish to start from the basic theory of C*-algebras and advance to an overview of some of the most spectacular results concerning the structure of nuclear C*-algebras.
The text is divided into three parts. First, elementary notions, classical theorems and constructions are developed. Then, essential examples in the theory, such as crossed products and the class of quasidiagonal C*-algebras, are examined, and finally, the Elliott invariant, the Cuntz semigroup, and the Jiang-Su algebra are defined. It is shown how these objects have played a fundamental role in understanding the fine structure of nuclear C*-algebras. To help understanding the theory, plenty of examples, treated in detail, are included.
This volume will also be valuable to researchers in the area as a reference guide. It contains an extensive reference list to guide readers that wish to travel further.

We extend the notion of Rokhlin dimension from topological dynamical systems to C∗-correspondences. We show that in the presence of finite Rokhlin dimension and a mild quasidiagonal-like condition (which, for example, is automatic for finitely generated projective correspondences), finite nuclear dimension passes from the scalar algebra to the associated Toeplitz–Pimsner and (hence) Cuntz–Pimsner algebras. As a consequence we provide new examples of classifiable C∗-algebras: if A is simple, unital, has finite nuclear dimension and satisfies the UCT, then for every finitely generated projective H with finite Rokhlin dimension, the associated Cuntz–Pimsner algebra O(H) is classifiable in the sense of Elliott’s Program.

I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the regularity conjecture and its interplay with internal and external approximation properties.

Let $A$ be a simple infinite dimensional stably finite unital C*-algebra, and let $B$ be a centrally large subalgebra of $A$. We prove that if $A$ is tracially ${\mathcal{Z}}$-absorbing if and only if $B$ is tracially ${\mathcal{Z}}$-absorbing. If $A$ and $B$ are also separable and nuclear, we prove that $A$ is ${\mathcal{Z}}$-absorbing if and only if $B$ is ${\mathcal{Z}}$-absorbing.

Let G and H be two locally compact groups acting on a C*-algebra A by commuting actions λ and σ. We construct an action on A XλG out of σ and a unitary cocycle u. For A commutative, and free and proper actions λ and σ, we show that if the roles of λ and σ are reversed, and u is replaced by u*, then the corresponding generalized fixed-point algebras, in the sense of Rieffel, are strong-Morita equivalent. This fact turns out to be a generalization of Green’s result on the strong-Morita equivalence of the algebras C0(M/H) XλG and C0(M/G) xσ H. Finally, we use the Morita equivalence mentioned above to compute the K-theory of quantum Heisenberg manifolds.

We prove that faithful traces on separable and nuclear C*-algebras in the UCT
class are quasidiagonal. This has a number of consequences. Firstly, by results
of many hands, the classification of unital, separable, simple and nuclear
C*-algebras of finite nuclear dimension which satisfy the UCT is now complete.
Secondly, our result links the finite to the general version of the Toms-Winter
conjecture in the expected way and hence clarifies the relation between
decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg
conjecture: discrete, amenable groups have quasidiagonal C*-algebras.

We define centrally large subalgebras of simple unital C*-algebras,
strengthening the definition of large subalgebras in previous work. We prove
that if A is any infinite dimensional simple separable unital C*-algebra which
contains a centrally large subalgebra with stable rank one, then A has stable
rank one. We also prove that large subalgebras of crossed product type are
automatically centrally large. We use these results to prove that if X is a
compact metric space which has a surjective continuous map to the Cantor set,
and h is a minimal homeomorphism of X, then C* (Z, X, h) has stable rank one,
regardless of the dimension of X or the mean dimension of h. In particular, the
Giol-Kerr examples give crossed products with stable rank one but which are not
stable under tensoring with the Jiang-Su algebra and are therefore not
classifiable in terms of the Elliott invariant.

The principal aim of this paper is to give a dynamical presentation of the
Jiang-Su algebra. Originally constructed as an inductive limit of prime
dimension drop algebras, the Jiang-Su algebra has gone from being a poorly
understood oddity to having a prominent positive role in George Elliott's
classification programme for separable, nuclear C*-algebras. Here, we exhibit
an etale equivalence relation whose groupoid C*-algebra is isomorphic to the
Jiang-Su algebra. The main ingredient is the construction of minimal
homeomorphisms on infinite, compact metric spaces, each having the same
cohomology as a point. This construction is also of interest in dynamical
systems. Any self-map of an infinite, compact space with the same cohomology as
a point has Lefschetz number one. Thus, if such a space were also to satisfy
some regularity hypothesis (which our examples do not), then the Lefschetz-Hopf
Theorem would imply that it does not admit a minimal homeomorphism.

This paper considers Hilbert C *-bimodules, a slight generalization of imprimitivity bimodules which were introduced by Rieffel [20]. Brown, Green, and Rieffel [7] showed that every imprimitivity bimodule X can be embedded into a certain C *-algebra L, called the linking algebra of X. We consider arbitrary embeddings of Hilbert C *-bimodules into C *-algebras; i.e. we describe the relative position of two arbitrary hereditary C *-algebras of a C *-algebra, in an analogy with Dixmier's description [10] of the relative position of two subspaces of a Hilbert space.
The main result of this paper (Theorem 4.3) is taken from the doctoral dissertation of the third author [22], although the proof here follows a different approach. In Section 1 we set out the definitions and basic properties (mostly folklore) of Hilbert C *-bimodules. In Section 2 we show how every quasi-multiplier gives rise to an embedding of a bimodule. In Section 3 we show that , the enveloping C *-algebra of the C *-algebraA with its product perturbed by a positive quasi-multiplier , is isomorphic to the closure (Proposition 3.1). Section 4 contains the main theorem (4.3), and in Section 5 we explain the analogy with the relative position of two subspaces of a Hilbert spaces and present some complements.

We consider the K-theory of C*-algebras of principal r-discrete groupoids. We describe two basic situations in which three groupoids are related; they can very loosely be described as "factor groupoids" and "sub-groupoids." For each, we show that there is a six-term exact sequence of associated K-groups. We present examples which arise from dynamical systems and from problems in the study of the orbit structure of topological systems. We also obtain the usual Mayer-Vietoris sequence in topological K-theory as a corollary.

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification. c 2012 by Princeton University Press. All Rights Reserved.

Suppose that A is a C*-algebra for which [Formula: see text], where [Formula: see text] is the Jiang–Su algebra: a unital, simple, stably finite, separable, nuclear, infinite-dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that:
(i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is almost unperforated.
(ii) If A is exact, then A is purely infinite if and only if A is traceless.
(iii) If A is separable and nuclear, then [Formula: see text] if and only if A is traceless.
(iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite.
We also characterize when A is of real rank zero.

Let \beta : S^n \to S^n, for n = 2k + 1, k \geq 1, be one of the known
examples of a non-uniquely ergodic minimal diffeomorphism of an odd dimensional
sphere. For every such minimal dynamical system (S^n, \beta) there is a Cantor
minimal system (X, \alpha) such that the corresponding product system (X x S^n,
\alpha x \beta) is minimal and the resulting crossed product C*-algebra C(X x
S^n) \rtimes_{\alpha x \beta} \mathbb{Z} is tracially approximately an interval
algebra (TAI). This entails classification for such C*-algebras. Moreover, the
minimal Cantor system (X, \alpha) is such that each tracial state on C(X x S^n)
\rtimes_{\beta} \mathbb{Z} induces the same state on the K_0-group and such
that the embedding of C(S^n) \rtimes_{\beta} \mathbb{Z} into C(X x S^n)
\rtimes_{\alpha x \beta} \mathbb{Z} preserves the tracial state space. This
implies C(S^n) \rtimes_{\beta} \mathbb{Z} is TAI after tensoring with the
universal UHF algebra, which in turn shows that the C*-algebras of these
examples of minimal diffeomorphisms of odd dimensional spheres are classified
by their tracial state spaces.

In this article I study a number of topological and algebraic dimension type properties of simple C∗-algebras and their interplay. In particular, a simple C∗-algebra is defined to be (tracially) \((m,\bar{m})\)-pure, if it has (strong tracial) m-comparison and is (tracially) \(\bar{m}\)-almost divisible. These notions are related to each other, and to nuclear dimension.
The main result says that if a separable, simple, nonelementary, unital C∗-algebra with locally finite nuclear dimension is \((m,\bar{m})\)-pure, then it absorbs the Jiang–Su algebra \(\mathcal{Z}\) tensorially. It follows that a separable, simple, nonelementary, unital C∗-algebra with locally finite nuclear dimension is \(\mathcal{Z}\)-stable if and only if it has the Cuntz semigroup of a \(\mathcal{Z}\)-stable C∗-algebra. The result may be regarded as a version of Kirchberg’s celebrated theorem that separable, simple, nuclear, purely infinite C∗-algebras absorb the Cuntz algebra \(\mathcal{O}_{\infty}\) tensorially.
As a corollary we obtain that finite nuclear dimension implies \(\mathcal{Z}\)-stability for separable, simple, nonelementary, unital C∗-algebras; this settles an important case of a conjecture by Toms and the author.
The main result also has a number of consequences for Elliott’s program to classify nuclear C∗-algebras by their K-theory data. In particular, it completes the classification of simple, unital, approximately homogeneous algebras with slow dimension growth by their Elliott invariants, a question left open in the Elliott–Gong–Li classification of simple AH algebras.
Another consequence is that for simple, unital, approximately subhomogeneous algebras, slow dimension growth and \(\mathcal {Z}\)-stability are equivalent. In the case where projections separate traces, this completes the classification of simple, unital, approximately subhomogeneous algebras with slow dimension growth by their ordered K-groups.

We study the topological variant of Rokhlin dimension for topological
dynamical systems (X,{\alpha},Z^m) in the case where X is assumed to have
finite covering dimension. Finite Rokhlin dimension in this sense is a property
that implies finite Rokhlin dimension of the induced action on C*-algebraic
level, as was discussed in a recent paper by Ilan Hirshberg, Wilhelm Winter and
Joachim Zacharias. In particular, it implies under these conditions that the
transformation group C*-algebra has finite nuclear dimension. Generalizing a
result of Yonatan Gutman, we show that free Z^m-actions on finite dimensional
spaces satisfy a strengthened version of the so-called marker property, which
yields finite Rokhlin dimension for said actions.

This corrects an error in the statement of Theorem 6 of my paper Vector Bundles and Projective Modules

Let X be an infinite compact metric space, \alpha : X \to X a minimal homeomorphism, u the unitary implementing \alpha in the transformation group C*-algebra, and S a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in S. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra Z, we prove that the transformation group C*-algebra tensored with a UHF algebra is tracially approximately S if there exists a y in X such that a certain C*-subalgebra is tracially approximately S. If the class S consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with Z for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either the transformation group C*-algebra or the C*-subalgebra, nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces. Comment: 20 pages

We develop a dilation theory for C*-correspondences, showing that every C*-correspondence E over a C*-algebra A can be universally embedded into a Hilbert C*-bimodule XE over a C*-algebra AE such that the crossed product A⋊E is naturally isomorphic to AE⋊XE . The Cuntz–Pimsner algebra E is isomorphic to E⋊E where E and E are quotients of AE, resp. XE. If E is full and the left action is by generalized compact operators, then E is an equivalence bimodule or, equivalently, an invertible C*-correspondence. In general, E is merely an essential Hilbert C*-bimodule. Slightly extending previous results on crossed products by equivalence bimodules, we apply our dilation theory to show that for full C*-correspondences over unital C*-algebras, E is simple if and only if E is minimal and nonperiodic, extending and simplifying results of Muhly and Solel and Kajiwara, Pinzari, and Watatani.

Let A be a simple unital C∗-algebra and let B be a UHF-algebra. We prove that the group of invertible elements in A ⊗ B is dense in A ⊗ B if A is stably finite, and that A ⊗ B is purely infinite otherwise. We give (partial) results on the size of the closure of the group of invertible elements in more general simple unital C∗-algebras.

We study C∗-algebras arising from C∗-correspondences, which were introduced by the author. We prove the gauge-invariant uniqueness theorem, and obtain conditions for our C∗-algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6-term exact sequence of K-groups involving the K-groups of our C∗-algebras.

We introduce the nuclear dimension of a C∗-algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive limits, tensor products, hereditary subalgebras (hence ideals), quotients, and even extensions. It can be computed for many examples; in particular, it is finite for all UCT Kirchberg algebras. In fact, all classes of nuclear C∗-algebras which have so far been successfully classified consist of examples with finite nuclear dimension, and it turns out that finite nuclear dimension implies many properties relevant for the classification program. Surprisingly, the concept is also linked to coarse geometry, since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform Roe algebra is dominated by the asymptotic dimension of the underlying space.

We introduce the completely positive rank, a notion of covering dimension for nuclear C∗-algebras and analyze some of its properties.The completely positive rank behaves nicely with respect to direct sums, quotients, ideals and inductive limits. For abelian C∗-algebras it coincides with covering dimension of the spectrum and there are similar results for continuous trace algebras.As it turns out, a C∗-algebra is zero-dimensional precisely if it is AF. We consider various examples, particularly of one-dimensional C∗-algebras, like the irrational rotation algebras, the Bunce–Deddens algebras or Blackadar's simple unital projectionless C∗-algebra.Finally, we compare the completely positive rank to other concepts of noncommutative covering dimension, such as stable or real rank.

We show that if A and B are C*–algebras which possess countable approximate identities, then A and B are stably isomorphic if and only if they are strongly Morita equivalent. By considering* Breuer ideals, we show that this may fail in the absence of countable approximate identities. Finally we discuss the Picard groups of C*–algebras, especially for stable algebras. © 1977, University of California, Berkeley. All Rights Reserved.

We investigate the structure of the C*-algebras associated with minimal homeomorphisms of the Cantor set via the crossed product construction. These C*-algebras exhibit many of the same properties as approximately finite dimensional (or AF) C*-algebras. Specifically, each non-empty closed subset of the Cantor set is shown to give rise, in a natural way, to an AF-subalgebra of the crossed product and we analyze these subalgebras. Results of Versik show that the crossed product may be embedded into an AF-algebra. We show that this embedding induces an order isomorphism at the level of Ko-groups. We examine examples arising from the theory of interval exchange transformations.

A C -algebra A is defined to be purely infinite if there are no characters on A, and if for every pair of positive elements a; b in A, such that b lies in the closed two-sided ideal generated by a, there exists a sequence fr n g in A such that r n ar n ! b. This definition agrees with the usual definition by J. Cuntz when A is simple. It is shown that the property of being purely infinite is preserved under extensions, Morita equivalence, inductive limits, and it passes to quotients, and to hereditary sub-C -algebras. It is shown that AOmega O1 is purely infinite for every C -algebra A. Purely infinite C -algebras admit no traces, and, conversely, an approximately divisible exact C -algebra is purely infinite if it admits no non-zero trace. 1 Introduction Joachim Cuntz introduced in [7] what is now called the Cuntz algebras O n (the universal C -algebra generated by n isometries whose range projections add up to the unit), and he showed that these C -algeb...

Let X be an infinite compact metric space with finite covering dimension and let h be a minimal homeomorphism of X. Let A be the associated crossed product C*-algebra. We show that A has tracial rank zero whenever the image of K_0 (A) in the affine functions on the tracial state space of A is dense. As a consequence, we show that these crossed product C*-algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values. By applying the classification theorem for nuclear simple C*-algebras with tracial rank zero, we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered K-theory.

We introduce a method to define $C^*$-algebras from $C^*$-correspondences. Our construction generalizes Cuntz-Pimsner algebras, crossed products by Hilbert $C^*$-modules, and graph algebras.

A study of Hilbert $C^*$-bimodules over commutative $C^*$-algebras is carried out and used to establish a sufficient condition for two quantum Heisenberg manifolds to be isomorphic. Comment: LaTex format, 19 pages, no figures

Cartan preserving isomorphisms between crossed products by Hilbert C(X)-bimodules

- M S Adamo
- M Forough
- K R Strung

M. S. Adamo, M. Forough, and K. R. Strung. Cartan preserving isomorphisms between
crossed products by Hilbert C(X)-bimodules. In preparation.

Nuclear dimension of simple C * -algebras

- J Castillejos
- S Evington
- A Tikuisis
- S White
- W Winter

J. Castillejos, S. Evington, A. Tikuisis, S. White, and W. Winter. Nuclear dimension of simple
C * -algebras. Invent. Math., 224(1):245-290, 2021.

C * -algebras by Example. Fields Institute Monographs

- K R Davidson

K. R. Davidson. C * -algebras by Example. Fields Institute Monographs. Amer. Math. Soc.,
Providence, R.I., 1996.

Classifiable C * -algebras from minimal Z-actions and their orbit-breaking subalgebras

- R J Deeley
- I F Putnam
- K R Strung

R. J. Deeley, I. F. Putnam, and K. R. Strung. Classifiable C * -algebras from minimal Z-actions
and their orbit-breaking subalgebras. arXiv preprint mathOA/2012.10947.