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

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.Rosický, and that the categories of unital (commutative) C * -algebras are not locally-isometry ℵ 0 -generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosický.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 ≤ 2. For the same reason, for a locally compact abelian group G the category of G-graded von Neumann algebras is not locally presentable.