Publication detail

Hilbert spaces and C*-algebras are not finitely concrete

LIEBERMAN, M. VASEY, S. ROSICKÝ, J.

English title

Hilbert spaces and C*-algebras are not finitely concrete

Type

journal article in Web of Science

Language

en

Original abstract

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. (C) 2022 Elsevier B.V. All rights reserved.

English abstract

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. (C) 2022 Elsevier B.V. All rights reserved.

Keywords in English

Hilbert space; C?-algebra; Faithful functor preserving directed  colimits

Released

01.04.2023

Publisher

ELSEVIER

Location

AMSTERDAM

ISSN

0022-4049

Volume

227

Number

4

Pages count

9

BIBTEX


@article{BUT181494,
  author="Michael Joseph {Lieberman} and Sebastien {Vasey} and Jiří {Rosický},
  title="Hilbert spaces and C*-algebras are not finitely concrete",
  year="2023",
  volume="227",
  number="4",
  month="April",
  publisher="ELSEVIER",
  address="AMSTERDAM",
  issn="0022-4049"
}