We study topological von Neumann regularity and principal von Neumann regularity of Banach algebras. Our main objective is comparing these two types of Banach algebras and some other known Banach algebras with one another. In particular, we show that the class of topologically von Neumann regular Banach algebras contains all $C^*$-algebras, group algebras of compact abelian groups and ...

Let H ⊴ G be a closed normal subgroup of locally compact quantum group. We introduce strictly positive group-like element affiliated with L ∞ ( ) that, roughly, measures the failure to act measure-preservingly on by conjugation. The triviality that is equivalent condition and / have same modular element, analogy classical situation. This automatic if ≤ central, in general implies unimodularity ...