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

Abstract

We study holomorphic maps between C${}^{\ast }$-algebras $A$ and $B$. When $f:B_A(0,\varrho )⟶B$ 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 $B^{\ast \ast }$ and Jordan ${}^{\ast }$-homomorphisms $\Theta ,\tilde{\Theta }:M(A)\to B^{\ast \ast }$ such that $$f(x)=\sum _{n=1}^{\infty }{h}_n\tilde{\Theta }(a^n)=\sum _{n=1}^{\infty }\Theta (a^n)h_n,$$ uniformly in $a\in U$. When $B$ is abelian the hypothesis of $B$ being unital and $f(U)\cap inv(B)\ne \varnothing$ can be relaxed to get the same statement.