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

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 $U=B_{A}(0,δ)$ 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 $Θ, \widetildeΘ : M(A) \to B^{**}$ such that $f(x) = \sum_{n=1}^\infty h_n \widetildeΘ (a^n)= \sum_{n=1}^\infty Θ (a^n) h_n,$ 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.

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