On preservers of strong Birkhoff-James orthogonality between $C^*$-algebras

TL;DR

This paper characterizes linear and nonlinear strong Birkhoff-James orthogonality preservers in unital and non-unital $C^*$-algebras, showing they are closely related to $*$-isomorphisms and providing a nonlinear characterization of compact $C^*$-algebras.

Contribution

It establishes that linear strong Birkhoff-James isomorphisms are essentially $*$-isomorphisms with unitary multiplication and extends this understanding to certain nonlinear maps in non-unital cases.

Findings

Linear strong Birkhoff-James isomorphisms are $*$-isomorphisms times a unitary.

Characterization of strong Birkhoff-James isomorphisms in compact $C^*$-algebras.

Nonlinear characterization of compact $C^*$-algebras via strong Birkhoff-James orthogonality.

Abstract

It is shown that every linear strong Birkhoff-James isomorphism between unital $C^*$-algebras is a $*$-isomorphism followed by a unitary multiplication. Moreover, as a partial extension of this result to the non-unital case, the form of (possibly nonlinear) strong Birkhoff-James isomorphisms between compact $C^*$-algebras are determined. A nonlinear characterization of compact $C^*$-algebras in terms of strong Birkhoff-James orthogonality is also given.