Monadic forgetful functors and (non-)presentability for $C^*$- and $W^*$-algebras

TL;DR

This paper investigates the categorical properties of $C^*$- and $W^*$-algebras, proving monadicity of certain forgetful functors and demonstrating the non-presentability of categories of von Neumann algebras.

Contribution

It establishes the monadicity of forgetful functors from $C^*$- and $W^*$-algebras to Banach structures and shows the non-presentability of von Neumann algebra categories.

Findings

Forgetful functors are monadic.

Categories of unital (commutative) $C^*$-algebras are not locally-isometry $eth_0$-generated.

Von Neumann algebra categories are not locally presentable; only 1-dimensional algebras are presentable.

Abstract

We prove that the forgetful functors from the categories of $C^*$- and $W^*$-algebras to Banach $*$-algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosick\'{y}, and that the categories of unital (commutative) $C^*$-algebras are not locally-isometry $\aleph_0$-generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosick\'{y}. We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are the two algebras of dimension $\le 1$. For the same reason, for a locally compact abelian group $\mathbb{G}$ the category of $\mathbb{G}$-graded von Neumann algebras is not locally presentable.