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1. CJM Online first

Cameron, Jan; Smith, Roger R.
A Galois correspondence for reduced crossed products of unital simple C$^*$-algebras by discrete groups
Let a discrete group $G$ act on a unital simple C$^*$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes_{\alpha,r}H$ between subgroups of $G$ and C$^*$-algebras $B$ satisfying $A\subseteq B \subseteq A\rtimes_{\alpha,r}G$, where $A\rtimes_{\alpha,r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\alpha,\sigma)$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes^\sigma_{\alpha,r}G$.

Keywords:C$^*$-algebra, group, crossed product, bimodule, reduced, twisted
Categories:46L55, 46L40

2. CJM 2013 (vol 65 pp. 783)

Garc├ęs, Jorge J.; Peralta, Antonio M.
Generalised Triple Homomorphisms and Derivations
We introduce generalised triple homomorphism between Jordan Banach triple systems as a concept which extends the notion of generalised homomorphism between Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively. We prove that every generalised triple homomorphism between JB$^*$-triples is automatically continuous. When particularised to C$^*$-algebras, we rediscover one of the main theorems established by B.E. Johnson. We shall also consider generalised triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$-module, proving that every generalised triple derivation from a JB$^*$-triple $E$ into itself or into $E^*$ is automatically continuous.

Keywords:generalised homomorphism, generalised triple homomorphism, generalised triple derivation, Banach algebra, Jordan Banach triple, C$^*$-algebra, JB$^*$-triple
Categories:46L05, 46L70, 47B48, 17C65, 46K70, 46L40, 47B47, 47B49

3. CJM 2006 (vol 58 pp. 1144)

Hamana, Masamichi
Partial $*$-Automorphisms, Normalizers, and Submodules in Monotone Complete $C^*$-Algebras
For monotone complete $C^*$-algebras $A\subset B$ with $A$ contained in $B$ as a monotone closed $C^*$-subalgebra, the relation $X = AsA$ gives a bijection between the set of all monotone closed linear subspaces $X$ of $B$ such that $AX + XA \subset X$ and $XX^* + X^*X \subset A$ and a set of certain partial isometries $s$ in the ``normalizer" of $A$ in $B$, and similarly for the map $s \mapsto \Ad s$ between the latter set and a set of certain ``partial $*$-automorphisms" of $A$. We introduce natural inverse semigroup structures in the set of such $X$'s and the set of partial $*$-automorphisms of $A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough $B$ the homomorphism becomes surjective and all the partial $*$-automorphisms of $A$ are realized via partial isometries in $B$. In particular, the inverse semigroup associated with a type ${\rm II}_1$ von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the $C^*$-algebra version of these results.

Categories:46L05, 46L08, 46L40, 20M18

4. CJM 2004 (vol 56 pp. 1237)

Kishimoto, Akitaka
Central Sequence Algebras of a Purely Infinite Simple $C^{*}$-algebra
We are concerned with a unital separable nuclear purely infinite simple $C^{*}$-algebra\ $A$ satisfying UCT with a Rohlin flow, as a continuation of~\cite{Kismh}. Our first result (which is independent of the Rohlin flow) is to characterize when two {\em central} projections in $A$ are equivalent by a {\em central} partial isometry. Our second result shows that the K-theory of the central sequence algebra $A'\cap A^\omega$ (for an $\omega\in \beta\N\setminus\N$) and its {\em fixed point} algebra under the flow are the same (incorporating the previous result). We will also complete and supplement the characterization result of the Rohlin property for flows stated in~ \cite{Kismh}.


5. CJM 2001 (vol 53 pp. 631)

Walters, Samuel G.
K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism
For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$ (containing the rationals) it is shown that the group $K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where $A_\theta$ is the rotation C*-algebra generated by unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) = U^{-1}$. More precisely, an explicit basis for $K_0$ consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital C*-algebras over $[0,1]$.) The Connes Chern character $\ch \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense $G_\delta$ set of parameters $\theta$. The main computational tool in this paper is a group homomorphism $\vtr \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$ obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of $H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$ is fully determined for each $\theta$. (We conjecture that this subspace is all of $H^{\ev}$.)

Keywords:C*-algebras, K-theory, automorphisms, rotation algebras, unbounded traces, Chern characters
Categories:46L80, 46L40, 19K14

6. CJM 2001 (vol 53 pp. 325)

Matui, Hiroki
Ext and OrderExt Classes of Certain Automorphisms of $C^*$-Algebras Arising from Cantor Minimal Systems
Giordano, Putnam and Skau showed that the transformation group $C^*$-algebra arising from a Cantor minimal system is an $AT$-algebra, and classified it by its $K$-theory. For approximately inner automorphisms that preserve $C(X)$, we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto's result.

Categories:46L40, 46L80, 54H20

7. CJM 2000 (vol 52 pp. 633)

Walters, Samuel G.
Chern Characters of Fourier Modules
Let $A_\theta$ denote the rotation algebra---the universal $C^\ast$-algebra generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where $\theta$ is a fixed real number. Let $\sigma$ denote the Fourier automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{-1}$, and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote the associated $C^\ast$-crossed product. It is shown that there is a canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each $\theta$ given by nine canonical modules. The unbounded trace functionals of $B_\theta$ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when $\theta$ is irrational. The Chern characters of the nine modules---and more importantly, the Fourier module---are computed and shown to involve techniques from the theory of Jacobi's theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense $G_\delta$ set of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is generated by the nine classes constructed here.

Keywords:$C^\ast$-algebras, unbounded traces, Chern characters, irrational rotation algebras, $K$-groups
Categories:46L80, 46L40

8. CJM 1999 (vol 51 pp. 850)

Muhly, Paul S.; Solel, Baruch
Tensor Algebras, Induced Representations, and the Wold Decomposition
Our objective in this sequel to \cite{MSp96a} is to develop extensions, to representations of tensor algebras over $C^{*}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling's theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras.

Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theorem
Categories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35

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