Hilbert spaces and C^∗-algebras are not finitely concrete

2019/08/27 by Michael Lieberman, Lieberman, Michael, Jiřı́ Rosický +3 · 1 citation
Computer Science · Mathematics · #18C35 (Primary) #46L05 #46M99 (Secondary) #Advanced Algebra and Logic #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1908.10200

openalex publication_date 2019/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that no faithful functor from the category of Hilbert spaces with linear isometries into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an analogous result for the category of commutative unital C^∗-algebras with ∗-homomorphisms. This implies, in particular, that this category is not axiomatizable by a first-order theory, a strengthening of a conjecture of Bankston.

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