Weakly Nuclear Maps on Real $C^*$-algebras and Quasidiagonality of These Algebras

TL;DR

This paper investigates the properties of weakly nuclear maps on real $C^*$-algebras, establishing their equivalence with complexifications, and explores the conditions under which real $C^*$-algebras are exact or quasidiagonal.

Contribution

It demonstrates that weak nuclearity, exactness, and quasidiagonality of real $C^*$-algebras correspond directly to their complexified counterparts, providing new insights into their structural properties.

Findings

Weakly nuclear maps on real $C^*$-algebras are equivalent to their complexifications.

A real $C^*$-algebra is exact if and only if its complexification is exact.

The same equivalence holds for quasidiagonality.

Abstract

In this paper, we show that a completely positive linear map is weakly nuclear if and only if its complexification is weakly nuclear. It is shown that a real $C^*$-algebra is exact if and only if its complexification is exact and similar case is provided for the quasidiaginality.