Hilbert spaces and C*-algebras are not finitely concrete
| Authors | |
|---|---|
| Year of publication | 2023 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Pure and Applied Algebra |
| MU Faculty or unit | |
| Citation | |
| Web | https://doi.org/10.1016/j.jpaa.2022.107245 |
| Doi | http://dx.doi.org/10.1016/j.jpaa.2022.107245 |
| 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. |
| Related projects: |