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Search: MSC category 46L40 ( Automorphisms )

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1. CMB Online first

Boutonnet, Remi; Roydor, Jean
A note on uniformly bounded cocycles into finite von Neumann algebras
We give a short proof of a result of T. Bates and T. Giordano stating that any uniformly bounded Borel cocycle into a finite von Neumann algebra is cohomologous to a unitary cocycle. We also point out a separability issue in their proof. Our approach is based on the existence of a non-positive curvature metric on the positive cone of a finite von Neumann algebra.

Keywords:Borel cocycle, von Neumann algebra
Categories:46L55, 46L40, 22D40

2. CMB 2006 (vol 49 pp. 371)

Floricel, Remus
Inner $E_0$-Semigroups on Infinite Factors
This paper is concerned with the structure of inner $E_0$-semigroups. We show that any inner $E_0$-semigroup acting on an infinite factor $M$ is completely determined by a continuous tensor product system of Hilbert spaces in $M$ and that the product system associated with an inner $E_0$-semigroup is a complete cocycle conjugacy invariant.

Keywords:von Neumann algebras, semigroups of endomorphisms, product systems, cocycle conjugacy
Categories:46L40, 46L55

3. CMB 2003 (vol 46 pp. 509)

Benson, David J.; Kumjian, Alex; Phillips, N. Christopher
Symmetries of Kirchberg Algebras
Let $G_0$ and $G_1$ be countable abelian groups. Let $\gamma_i$ be an automorphism of $G_i$ of order two. Then there exists a unital Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and with $[1_A] = 0$ in $K_0 (A)$, and an automorphism $\alpha \in \Aut(A)$ of order two, such that $K_0 (A) \cong G_0$, such that $K_1 (A) \cong G_1$, and such that $\alpha_* \colon K_i (A) \to K_i (A)$ is $\gamma_i$. As a consequence, we prove that every $\mathbb{Z}_2$-graded countable module over the representation ring $R (\mathbb{Z}_2)$ of $\mathbb{Z}_2$ is isomorphic to the equivariant $K$-theory $K^{\mathbb{Z}_2} (A)$ for some action of $\mathbb{Z}_2$ on a unital Kirchberg algebra~$A$. Along the way, we prove that every not necessarily finitely generated $\mathbb{Z} [\mathbb{Z}_2]$-module which is free as a $\mathbb{Z}$-module has a direct sum decomposition with only three kinds of summands, namely $\mathbb{Z} [\mathbb{Z}_2]$ itself and $\mathbb{Z}$ on which the nontrivial element of $\mathbb{Z}_2$ acts either trivially or by multiplication by $-1$.

Categories:20C10, 46L55, 19K99, 19L47, 46L40, 46L80

4. CMB 2003 (vol 46 pp. 365)

Kishimoto, Akitaka; Ozawa, Narutaka; Sakai, Shôichirô
Homogeneity of the Pure State Space of a Separable $C^*$-Algebra
We prove that the pure state space is homogeneous under the action of the automorphism group (or the subgroup of asymptotically inner automorphisms) for all the separable simple $C^*$-algebras. The first result of this kind was shown by Powers for the UHF algbras some 30 years ago.

Categories:46L40, 46L30

5. CMB 2001 (vol 44 pp. 335)

Stacey, P. J.
Inductive Limit Toral Automorphisms of Irrational Rotation Algebras
Irrational rotation $C^*$-algebras have an inductive limit decomposition in terms of matrix algebras over the space of continuous functions on the circle and this decomposition can be chosen to be invariant under the flip automorphism. It is shown that the flip is essentially the only toral automorphism with this property.

Categories:46L40, 46L35

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