Hyperreflexivity of von Neumann algebras and similarity of finitely generated $C^*$-algebras

TL;DR

This work studies Kadison's Similarity Problem for $C^*$-algebras by introducing projection-extension hypotheses (EPH.1--EPH.3) that tie SP to hyperreflexivity of von Neumann algebras and their commutants. It proves EPH.1 is equivalent to universal hyperreflexivity (and thus SP for all $C^*$-algebras), and that EPH.2 implies EPH.3, linking finite generation to SP via completely hyperreflexive structures and matrix amplifications. It further connects weak similarity properties (WSP) for von Neumann algebras to the finite-generation and single-generation problems, showing that SP for finitely generated $C^*$-algebras is equivalent to WSP under certain generation hypotheses. Collectively, the results provide a structured route where preserving hyperreflexivity under simple projection extensions suffices to resolve SP for broad classes, and they reveal deep connections between finite generation, projections, and Kadison's open problems.

Abstract

Let $A$ be a $C^*$-algebra. We say that $A$ satisfies the SP if every bounded homomorphism $A\to B(K)$, with $K$ a Hilbert space, is similar to a $*$-homomorphism. We introduce three hypotheses that relate to extending hyperreflexive algebras by projections. We prove that our third hypothesis is equivalent to every finitely generated C*-algebra satisfying the SP. We show that to prove that every von Neumann algebra is hyperreflexive it is enough to show that when one extends a hyperreflexive algebra by a single projection it remains hyperreflexive.

Theorems & Definitions (60)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Remark 1.4
• Definition 1.5
• Theorem 1.6
• Corollary 1.7
• proof
• Definition 1.8
• Definition 1.9