A reflexivity problem concerning the $C^*$-algebra $C(X)\otimes B(H)$

The principle of local reflexivity and an extension of the identity $\mathcal B(E,X^{**})\cong\mathcal B(E,X)^{**}$

By using the Principle of Local Reflexivity (PLR), we prove that for every two Banach spaces $E$ and $X$ there exists a suitable ultrafilter $\mathcal{U}$ such that $ \mathcal{F}(E,X)^*,$ the dual space of the finite rank operators, can be isomorphically identified with certain quotient of the ultrapower space $(E\widehat{\otimes} X^*)_\mathcal{U}$, of the projective tensor product space $E\widehat{\otimes} X^*.$ This generalizes the identity $\mathcal B(E,X^{**})\cong\mathcal B(E,X)^{**}$, where $E$ is finite-dimensional. We then serve our main result to improve some results on the reflexivit…

Semicrossed Products and Reflexivity

Given a w*-closed unital algebra $A$ acting on $H_0$ and a contractive w*-continuous endomorphism $β$ of $A$, there is a w*-closed (non-selfadjoint) unital algebra $\mathbb{Z}_+\bar{\times}_βA$ acting on $H_0\otimes\ell^2({\mathbb{Z}_+})$, called the w*-semicrossed product of $A$ with $β$. We prove that the w*-semicrossed product is a reflexive operator algebra provided $A$ is reflexive and $β$ is unitarily implemented, and that it has the bicommutant property if and only if so does $A$. Also, we show that the w*-semicrossed product generated by a commutative C*-algebra and a *-endomorphism is…

Operator algebras from the discrete Heisenberg semigroup

We study reflexivity and structure properties of operator algebras generated by representations of the discrete Heisenberg semi-group. We show that the left regular representation of this semi-group gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation which gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of $H^{\infty}(\bb{T})\otimes\cl B(\cl H)$.

Multiplicative maps on ideals of operators which are local automorphisms

We present the following reflexivity-like result concerning the automorphism group of the $C^*$-algebra B(H), H being a separable Hilbert space. Let $ϕ:B(H)\to B(H)$ be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of B(H) (these automorphisms may, of course, depend on the point) in the operator norm. Then $ϕ$ is an automorphism of the algebra B(H).

Simultaneous similarity, bounded generation and amenability

We prove that a discrete group $G$ is amenable iff it is strongly unitarizable in the following sense: every unitarizable representation $π$ on $G$ can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of $π$. Analogously a $C^*$-algebra $A$ is nuclear iff any bounded homomorphism $u: A\to B(H)$ is strongly similar to a $*$-homomorphism in the sense that there is an invertible operator $ξ$ in the von Neumann algebra generated by the range of $u$ such that $a\to ξu(a) ξ^{-1}$ is a $*$-homomorphism. An analogous characterization holds in terms of derivations…

关于与 Abel von Neumann 代数张量积相关的主题

On Topics Related to Tensor Products with Abelian von Neumann Algebras

本文研究 Abel von Neumann 代数的谱及其张量积，通过 Boolean 代数超滤刻画谱、定义迹超幂，并给出适用于非可分情形的连续函数表示及 $\mathrm{II}_1$ 因子同构结构结果。