论文标题
$ c^*$ - 和$ w^*$ - 代数
Monadic forgetful functors and (non-)presentability for $C^*$- and $W^*$-algebras
论文作者
论文摘要
我们证明，$ c^*$ - 和$ w^*$ - 代数为Banach $*$的代数 - 代数，Banach代数或Banach空间的代数都是单声道的，回答了J.rosický的问题，回答了J.rosický的问题，而不是c^*$ c^*$ - algebras的类别 - 作为富含公制的类别，回答了I. Di Liberti和Rosický的问题。 我们还证明了von Neumann代数类别的许多负面性结果：不仅该类别不可本地呈现，而且实际上它的唯一可呈现的对象是维度$ \ le 1 $的两个代数。出于同样的原因，对于本地紧凑的Abelian组$ \ Mathbb {g} $ $ \ Mathbb {G} $的类别 - 分级的Von Neumann代数分级并非本地提供。
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 $\aleph_0$-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 $\le 1$. 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.
