Orthogonally additive, orthogonality preserving, holomorphic mappings between C*-algebras

TL;DR

This paper characterizes holomorphic maps between C*-algebras that preserve orthogonality and are orthogonally additive, providing a detailed structure involving Jordan *-homomorphisms and elements in the bidual.

Contribution

It establishes a structural representation for orthogonality preserving, orthogonally additive holomorphic maps between C*-algebras, extending previous understanding of such mappings.

Findings

Holomorphic maps can be expressed via Jordan *-homomorphisms and elements in the bidual.

Orthogonality preservation implies a specific algebraic structure for the maps.

Results hold under relaxed conditions when the target algebra is abelian.

Abstract

We study holomorphic maps between C$^*$-algebras $A$ and $B$. When $f:B_A (0,\varrho) \longrightarrow B$ is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball $U=B_{A}(0,\delta)$ and we assume that $f$ is orthogonality preserving on $A_{sa}\cap U$, orthogonally additive on $U$ and $f(U)$ contains an invertible element in $B$, then there exist a sequence $(h_n)$ in $B^{**}$ and Jordan $^*$-homomorphisms $\Theta, \widetilde{\Theta} : M(A) \to B^{**}$ such that $$ f(x) = \sum_{n=1}^\infty h_n \widetilde{\Theta} (a^n)= \sum_{n=1}^\infty {\Theta} (a^n) h_n,$$ uniformly in $a\in U$. When $B$ is abelian the hypothesis of $B$ being unital and $f(U)\cap \hbox{inv} (B) \neq \emptyset$ can be relaxed to get the same statement.