Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras

Abstract

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