Publication details

Hilbert spaces and C*-algebras are not finitely concrete

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LIEBERMAN Michael Joseph ROSICKÝ Jiří VASEY Sébastien Bernard

Year of publication 2023
Type Article in Periodical
Magazine / Source Journal of Pure and Applied Algebra
MU Faculty or unit

Faculty of Science

Keywords Hilbert space; C*-algebra; Faithful functor preserving directed; colimits
Description We show that no faithful functor from the category of Hilbert spaces with injective linear contractions 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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