On preservers of strong Birkhoff-James orthogonality between $C^*$-algebras
Bojan Kuzma, Srdjan Stefanović, Ryotaro Tanaka
TL;DR
The paper provides a tight structural classification of preservers of strong Birkhoff-James orthogonality between $C^*$-algebras. It shows that every linear strong BJ isomorphism between unital algebras is, up to a scalar factor, a $*$-isomorphism composed with a unitary multiplier, and it extends partial nonlinear classifications to compact algebras via a block-decomposition framework. A complete nonlinear characterization of compact $C^*$-algebras by strong BJ orthogonality is given, together with a precise description of nonlinear and linear preservers on the compact operators $K( rak H)$, including wild nonlinear examples in higher dimensions. The results connect deep operator-theoretic concepts (multiplier algebras, Jordan $*$-isomorphisms, and block structure) to a robust preservation phenomenon, thereby elucidating how the metric/Jordan structure of a $C^*$-algebra encodes its full algebraic form. Significantly, the work reduces the linear strong BJ preservers to isometric maps induced by $*$-isomorphisms and unitary conjugations, while exposing a rich nonlinear landscape in the compact setting.
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.