The symbiosis of C*- and W*-algebras

TL;DR

This paper reviews the deep connections between C*-algebras and von Neumann algebras, emphasizing the importance of understanding both to advance operator algebra theory.

Contribution

It highlights the historical and mathematical interplay between C*- and W*-algebras, advocating for integrated knowledge to foster progress in the field.

Findings

Connections between C*- and von Neumann algebras are fundamental to operator algebra theory.

Understanding both algebra types enhances insights into amenability.

Interplay between the two theories has led to significant mathematical results.

Abstract

These days it is common for young operator algebraists to know a lot about C*-algebras, or a lot about von Neumann algebras -- but not both. Though a natural consequence of the breadth and depth of each subject, this is unfortunate as the interplay between the two theories has deep historical roots and has led to many beautiful results. We review some of these connections, in the context of amenability, with the hope of convincing (younger) readers that tribalism impedes progress.