2023/06/13 by Eleftherakis, G. K., Papapetros, E. · 1 citation
#FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2306.07605
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→ \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 w^*-conitnuous unital and bounded homomorphism u\colon \cl A→ \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′ ⊗ \cl B(ℓ2(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^′ ⊗ \cl B(ℓ2(\bbN)) 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 ⊗ \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 w^*-similarity degree d*(\cl A ⊗ \cl B).