The unitary group in the strong topology and a construction of Dixmier-Douady

2025/04/15 by Nikolai V. Ivanov, Ivanov, Nikolai V., Marina L. Prokhorova +1
Mathematics · #46L80 #47A99 (Secondary) #47D03 #57S05 (Primary) 19K56 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2504.11646

openalex publication_date 2025/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By a theorem of Dixmier-Douady the unitary group of an infinite-dimensional separable Hilbert space H in the strong operator topology is contractible. The Dixmier-Douady proof is based on an explicit construction of families of subspaces and operators in H with rather special properties. Unfortunately, this proof leaves hidden the geometric meaning of the theorem. The first goal of this note is to give a direct geometric proof of this theorem. The second goal is to provide a geometic analogue of Dixmier-Douady construction.

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