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Search: MSC category 46L55 ( Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20] )

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1. CJM Online first

Osaka, Hiroyuki; Teruya, Tamotsu
The Jiang-Su absorption for inclusions of unital C*-algebras
We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital C*-algebras $P \subset A$ with index finite, and show that an action $\alpha$ from a finite group $G$ on a simple unital C*-algebra $A$ has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation $E\colon A \rightarrow A^G$ has the tracial Rokhlin property. Let $\mathcal{C}$ be a class of infinite dimensional stably finite separable unital C*-algebras which is closed under the following conditions: (1) If $A \in {\mathcal C}$ and $B \cong A$, then $B \in \mathcal{C}$. (2) If $A \in \mathcal{C}$ and $n \in \mathbb{N}$, then $M_n(A) \in \mathcal{C}$. (3) If $A \in \mathcal{C}$ and $p \in A$ is a nonzero projection, then $pAp \in \mathcal{C}$. Suppose that any C*-algebra in $\mathcal{C}$ is weakly semiprojective. We prove that if $A$ is a local tracial $\mathcal{C}$-algebra in the sense of Fan and Fang and a conditional expectation $E\colon A \rightarrow P$ is of index-finite type with the tracial Rokhlin property, then $P$ is a unital local tracial $\mathcal{C}$-algebra. The main result is that if $A$ is simple, separable, unital nuclear, Jiang-Su absorbing and $E\colon A \rightarrow P$ has the tracial Rokhlin property, then $P$ is Jiang-Su absorbing. As an application, when an action $\alpha$ from a finite group $G$ on a simple unital C*-algebra $A$ has the tracial Rokhlin property, then for any subgroup $H$ of $G$ the fixed point algebra $A^H$ and the crossed product algebra $A \rtimes_{\alpha_{|H}} H$ is Jiang-Su absorbing. We also show that the strict comparison property for a Cuntz semigroup $W(A)$ is hereditary to $W(P)$ if $A$ is simple, separable, exact, unital, and $E\colon A \rightarrow P$ has the tracial Rokhlin property.

Keywords:Jiang-Su absorption, inclusion of C*-algebra, strict comparison
Categories:46L55, 46L35

2. CJM Online first

Eilers, Søren; Restorff, Gunnar; Ruiz, Efren; Sørensen, Adam P. W.
Geometric classification of graph C*-algebras over finite graphs
We address the classification problem for graph $C^*$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph $C^*$-algebras may come with uncountably many ideals. We find that in this generality, stable isomorphism of graph $C^*$-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the $C^*$-algebras having isomorphic $K$-theories. This proves in turn that under this condition, the graph $C^*$-algebras are in fact classifiable by $K$-theory, providing in particular complete classification when the $C^*$-algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs. Our results are applied to discuss the classification problem for the quantum lens spaces defined by Hong and Szymański, and to complete the classification of graph $C^*$-algebras associated to all simple graphs with four vertices or less.

Keywords:graph $C^*$-algebra, geometric classification, $K$-theory, flow equivalence
Categories:46L35, 46L80, 46L55, 37B10

3. CJM 2016 (vol 68 pp. 1023)

Phillips, John; Raeburn, Iain
Centre-valued Index for Toeplitz Operators with Noncommuting Symbols
We formulate and prove a ``winding number'' index theorem for certain ``Toeplitz'' operators in the same spirit as Gohberg-Krein, Lesch and others. The ``number'' is replaced by a self-adjoint operator in a subalgebra $Z\subseteq Z(A)$ of a unital $C^*$-algebra, $A$. We assume a faithful $Z$-valued trace $\tau$ on $A$ left invariant under an action $\alpha:{\mathbf R}\to Aut(A)$ leaving $Z$ pointwise fixed.If $\delta$ is the infinitesimal generator of $\alpha$ and $u$ is invertible in $\operatorname{dom}(\delta)$ then the ``winding operator'' of $u$ is $\frac{1}{2\pi i}\tau(\delta(u)u^{-1})\in Z_{sa}.$ By a careful choice of representations we extend $(A,Z,\tau,\alpha)$ to a von Neumann setting $(\mathfrak{A},\mathfrak{Z},\bar\tau,\bar\alpha)$ where $\mathfrak{A}=A^{\prime\prime}$ and $\mathfrak{Z}=Z^{\prime\prime}.$ Then $A\subset\mathfrak{A}\subset \mathfrak{A}\rtimes{\bf R}$, the von Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$-trace on $\mathfrak{A}\rtimes{\bf R}$. If $P$ is the projection in $\mathfrak{A}\rtimes{\bf R}$ corresponding to the non-negative spectrum of the generator of $\mathbf R$ inside $\mathfrak{A}\rtimes{\mathbf R}$ and $\tilde\pi:A\to\mathfrak{A}\rtimes{\mathbf R}$ is the embedding then we define for $u\in A^{-1}$, $T_u=P\tilde\pi(u) P$ and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$-valued index of $T_u$ is the negative of the winding operator. In outline the proof follows the proof of the scalar case done previously by the authors. The main difficulty is making sense of the constructions with the scalars replaced by $\mathfrak{Z}$ in the von Neumann setting. The construction of the dual $\mathfrak{Z}$-trace on $\mathfrak{A}\rtimes{\mathbf R}$ required the nontrivial development of a $\mathfrak{Z}$-Hilbert Algebra theory. We show that certain of these Fredholm operators fiber as a ``section'' of Fredholm operators with scalar-valued index and the centre-valued index fibers as a section of the scalar-valued indices.

Keywords:index ,Toeplitz operator
Categories:46L55, 19K56, 46L80

4. CJM 2015 (vol 67 pp. 481)

an Huef, Astrid; Archbold, Robert John
The C*-algebras of Compact Transformation Groups
We investigate the representation theory of the crossed-product $C^*$-algebra associated to a compact group $G$ acting on a locally compact space $X$ when the stability subgroups vary discontinuously. Our main result applies when $G$ has a principal stability subgroup or $X$ is locally of finite $G$-orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation $V$ of a stability subgroup is obtained by restricting $V$ to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of $V$. As a corollary we obtain that when the trivial subgroup is a principal stability subgroup, the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the $C^*$-algebra of the motion group $\mathbb{R}^n\rtimes \operatorname{SO}(n)$ is a Fell algebra. This uses the classical branching theorem for the special orthogonal group $\operatorname{SO}(n)$ with respect to $\operatorname{SO}(n-1)$. Since proper transformation groups are locally induced from the actions of compact groups, we describe how some of our results can be extended to transformation groups that are locally proper.

Keywords:compact transformation group, proper action, spectrum of a C*-algebra, multiplicity of a representation, crossed-product C*-algebra, continuous-trace C*-algebra, Fell algebra
Categories:46L05, 46L55

5. CJM 2008 (vol 60 pp. 975)

Boca, Florin P.
An AF Algebra Associated with the Farey Tessellation
We associate with the Farey tessellation of the upper half-plane an AF algebra $\AA$ encoding the ``cutting sequences'' that define vertical geodesics. The Effros--Shen AF algebras arise as quotients of $\AA$. Using the path algebra model for AF algebras we construct, for each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in $\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.

Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20

6. CJM 2006 (vol 58 pp. 39)

Exel, R.; Vershik, A.
$C^*$-Algebras of Irreversible Dynamical Systems
We show that certain $C^*$-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed-product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.

Categories:46L55, 37A55

7. CJM 2005 (vol 57 pp. 983)

an Huef, Astrid; Raeburn, Iain; Williams, Dana P.
A Symmetric Imprimitivity Theorem for Commuting Proper Actions
We prove a symmetric imprimitivity theorem for commuting proper actions of locally compact groups $H$ and $K$ on a $C^*$-algebra.

Categories:46L05, 46L08, 46L55

8. CJM 2003 (vol 55 pp. 1302)

Katsura, Takeshi
The Ideal Structures of Crossed Products of Cuntz Algebras by Quasi-Free Actions of Abelian Groups
We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A.~Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and $K$-groups of our algebras.

Categories:46L05, 46L55, 46L45

9. CJM 1999 (vol 51 pp. 745)

Echterhoff, Siegfried; Quigg, John
Induced Coactions of Discrete Groups on $C^*$-Algebras
Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a $C^*$-coaction $\delta\colon D\to D\otimes C^*(G/N)$ of a quotient group $G/N$ of a discrete group $G$ to a $C^*$-coaction $\Ind\delta\colon\Ind D\to \Ind D\otimes C^*(G)$ of $G$. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products $\Ind D\times_{\Ind\delta}G$ and $D\times_{\delta}G/N$ are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.


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