On C*-algebras which cannot be decomposed into tensor products with both factors infinite-dimensional

TL;DR

This paper demonstrates that certain C*-algebras, specifically those that are Grothendieck as Banach spaces, cannot be expressed as tensor products of two infinite-dimensional C*-algebras, expanding understanding of their structural limitations.

Contribution

It establishes a new class of C*-algebras that cannot be decomposed into tensor products with both factors infinite-dimensional, generalizing previous results to Grothendieck C*-algebras.

Findings

Grothendieck C*-algebras cannot be decomposed into tensor products of two infinite-dimensional C*-algebras

Includes all von Neumann algebras and their norm-quotients as a subclass

Complements recent results on SAW*-algebras by Ghasemi.

Abstract

We prove that C*-algebras which, as Banach spaces, are Grothendieck cannot be decomposed into a tensor product of two infinite-dimensional C*-algebras. By a result of Pfitzner, this class contains all von Neumann algebras and their norm-quotients. We thus complement a recent result of Ghasemi who established a similar conclusion for the class of SAW*-algebras.