Journal of Operator Theory

Volume 94, Issue 1, Summer 2025 pp. 195-217.

The similarity problem and hyperreflexivity of von Neumann algebras

Authors: G.K. Eleftherakis (1), E. Papapetros (2)
Author institution: (1) Department of Mathematics, University of Patras, 265 00 Patras, Greece
(2) Department of Mathematics, University of Patras, 265 00 Patras, Greece

Summary: We say that a $C^*$-algebra $\mathcal A$ satisfies the similarity property ((SP)) if every bounded homomorphism $u\colon \mathcal A\to \mathcal B(H),$ where $H$ is a Hilbert space, is similar to a $*$-homomorphism and that a von Neumann algebra $\mathcal A$ satisfies the weak similarity property ((WSP)) if every $\mathrm{w}^*$-continuous, unital and bounded homomorphism $u\colon \mathcal A\to \mathcal B(H),$ where $H$ is a Hilbert space, is similar to a $*$-homomorphism. We prove that a von Neumann algebra $\mathcal A$ satisfies (WSP) if and only if the algebras $\mathcal A^{\prime}\overline \otimes \mathcal B(\ell^2(J))$ are hyperreflexive for all cardinals $J.$ We also introduce the hypothesis (CHH): every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under (CHH), all $C^*$-algebras satisfy (SP). Finally, we prove that the spatial tensor product of an injective von Neumann algebra and a von Neumann algebra satisfying (WSP) also satisfies (WSP).

DOI: http://dx.doi.org/10.7900/jot.2023oct13.2441
Keywords: von Neumann algebras, similarity, hyperreflexivity, injectivity

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