Hilbert spaces and ${C}^\ast$-algebras are not finitely concrete

Michael Lieberman; Jiří Rosický; Sebastien Vasey

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arXiv · 1908.10200

Hilbert spaces and ${C}^\ast$-algebras are not finitely concrete

Michael Lieberman·Jiří Rosický·Sebastien Vasey

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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 $C^\ast$-algebras with $\ast$-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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BibTeX RIS

Michael Lieberman, Jiří Rosický, Sebastien Vasey. 2019-08-27. Hilbert spaces and ${C}^\ast$-algebras are not finitely concrete. https://arxiv.org/abs/1908.10200

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