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October 16, 2025Open Access

Almost Representations

Key Points

The main finding shows a correspondence between maps from amenable groups to unitary groups and specific bounded linear operators.
Proven for any epsilon > 0, finite subsets, and bounded operators meets the structure, revealing an unexpected depth in operator relationships.
Approach employs properties of finite subsets of groups and linear maps in the context of separable infinite dimensional Hilbert spaces.
This result indicates limits to the fullness condition, calling for a deeper exploration of amenable groups in operator algebras.

Abstract

Let H be an infinite dimensional separable Hilbert space, B (H) the C^*-algebra of all bounded linear operators on H, U (B (H) ) the unitary group of B (H) and K B (H) the ideal of compact operators. Let G be a countable discrete amenable group. We prove the following: For any >0, any finite subset F G, and 00, finite subsets G G and S CG satisfying the following property: For any map: G U (B (H) ) such that \| (fg) - (f) (g) \|<\, \, \, for\, \, all\, \, f, g G\, \, \, and \, \, \, \| (x) \| \|x\|\, \, \, for\, \, all\, \, x S, there is a group homomorphism h: G U (B (H) ) such that \| (f) -h (f) \|<\, \, \, for\, \, \, all\, \, \, f F, where is the linear extension of on the group ring CG and: B (H) B (H) / K is the quotient map. A counterexample is given that the fullness condition above cannot be removed. We actually prove a more general result for separable amenable C^*-algebras.

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Cite This Study

Huaxin Lin (Mon,) studied this question.

synapsesocial.com/papers/68f0d5eb105731330a2b2006 https://doi.org/https://doi.org/10.48550/arxiv.2502.17802

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