arXiv subjects
Publications and source records attributed to E. Papapetros.
We say that a $C^*$-algebra $\mathcal{A}$ satisfies the similarity property ((SP)) if every bounded homomorphism $u\colon \mathcal{A} \to \mathcal{B}(\mathit{H})$, where $\mathit{H}$ is a Hilbert space, is similar to a $*$-homomorphism. We introduce the following hypothesis (EP). (EP): Every separably acting von Neumann algebra with a cyclic vector is hyperreflexive. We prove that under (EP), all $C^*$-algebras satisfy (SP).
The similarity problem is one of the most famous open problems in the theory of $C^*$-algebras. We say that a $C^*$-algebra $\cl A$ satisfies the similarity property ((SP) for short) if every bounded homomorphism $u\colon \cl A\to \cl B(H)$ is similar to a $*$-homomorphism and that a von Neumann algebra $\cl A$ satisfies the weak similarity property ((WSP) for short) if every $\mathrm{w}^*$-conitnuous unital and bounded homomorphism $u\colon \cl A\to \cl B(H),$ where $H$ is a Hilbert space, is similar to a $*$-homomorphism. We prove that a von Neumann algebra $\cl A$ satisfies (WSP) if and only if the algebras $\cl A^{\prime}\bar \otimes \cl B(\ell^2(I))$ are hyperreflexive for all cardinals $I.$ In the case in which $\cl A$ is a separably acting von Neumann algebra we prove that it satisfies (WSP) if and only if the algebra $\cl A^\prime \bar \otimes \cl B(\ell^2(\bb{N}))$ is hyperreflexive. We also introduce the hypothesis {\bf (CHH)}: Every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under {\bf (CHH)}, all $C^*$-algebras satisfy (SP). Finally, we prove that the spatial tensor product $\cl A\bar \otimes \cl B,$ where $\cl A$ is an injective von Neumann algebra and $\cl B$ is a von Neumann algebra satisfying (WSP), also satisfies (WSP) and we provide an upper bound for the $\text{w}^*$-similarity degree $d_{*}(\cl A\bar \otimes \cl B).$
Rigged modules over an operator algebra are a generalization of Hilbert modules over a $C^{\star}$-algebra. We characterize the rigged modules over an operator algebra $\mathcal A$ which are orthogonally complemented in $C_\infty(\mathcal A),$ the space of infinite columns with entries in $\mathcal A.$ We show that every such rigged module `restricts' to a bimodule of Morita equivalence between appropriate stably isomorphic operator algebras.
We introduce the notion of $Δ$ and $σ\,Δ-$ pairs for operator algebras and characterise $Δ-$ pairs through their categories of left operator modules over these algebras. Furthermore, we introduce the notion of $Δ$-Morita equivalent operator spaces and prove a similar theorem about their algebraic extensions. We prove that $σΔ$-Morita equivalent operator spaces are stably isomorphic and vice versa. Finally, we study unital operator spaces, emphasising their left (resp. right) multiplier algebras, and prove theorems that refer to $Δ$-Morita equivalence of their algebraic extensions.