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October 16, 2025Open Access

The unitary group of a II₁ factor is SOT-contractible

Key Points

The unitary group of any SOT-separable II1 factor is contractible, impacting the structure of these mathematical objects.
This finding extends to SOT-separable von Neumann algebras without type I_n direct summands, enriching algebraic theory.
The proof employs techniques like free convolution and Popa's theorem, showcasing the interplay between different mathematical concepts.
Understanding these contractibility results may influence future research in operator algebras and their applications.

Abstract

We show that the unitary group of any SOT-separable II₁ factor M, with the strong operator topology, is contractible. Combined with several old results, this implies that the same is true for any SOT-separable von Neumann algebra with no type Iₙ direct summands (n <). The proof for the II₁-factor case uses regularization via free convolution and Popa's theorem on the existence of approximately free Haar unitaries in II₁ factors.

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Cite This Study

David Jekel (Thu,) studied this question.

synapsesocial.com/papers/68f0f51d8dd8ea469b1d713e https://doi.org/https://doi.org/10.48550/arxiv.2508.05834

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