## Orthogonally additive holomorphic maps between C$^{*}$-algebras

### Volume 234 / 2016

#### Abstract

Let $A,B$ be $\rm C^{*}$-algebras, $B_A(0;r)$ the open ball in $A$ centered at $0$ with radius $r \gt 0$, and $H:B_A(0;r)\to B$ an orthogonally additive holomorphic map. If $H$ is zero product preserving on positive elements in $B_A(0;r)$, we show, in the commutative case, i.e., $A=C_0(X)$ and $B=C_0(Y)$, that there exist weight functions $h_n$ and a symbol map $\varphi : Y\to X$ such that $$ H(f)=\sum _{n\geq 1} h_n (f\circ \varphi )^n, \hskip 1em \ \forall f\in B_{C_0(X)}(0;r). $$ In the general case, we show that if $H$ is also conformal then there exist central multipliers $h_n$ of $B$ and a surjective Jordan isomorphism $J: A\to B$ such that $$ H(a) = \sum _{n\geq 1} h_n J(a)^n, \hskip 1em\ \forall a\in B_A(0;r). $$ If, in addition, $H$ is zero product preserving on the whole $B_A(0;r)$, then $J$ is an algebra isomorphism.

We also study orthogonally additive $n$-homogeneous polynomials which are $n$-isometries.