Orthogonally additive holomorphic maps between C*-algebras

Let $A,B$ be C*-algebras, $B_A(0;r)$ the open ball in $A$ centered at $0$ with radius $r>0$, 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 when $A=C_0(X)$ and $B=C_0(Y)$, that there exist weight functions $h_n$’s and a symbol map $\phi :Y\to X$ such that $$H(f)=\sum _{n\ge 1}{h}_n(f\circ \phi {)}^n,\mathrm{\forall }f\in B_{C_0(X)}(0;r)\mathrm{.}$$ In the general case, we show that if $H$ is also conformal then there exist central multipliers $h_n$’s of $B$ and a surjective Jordan isomorphism $J: A\to B$suchthat$$H(a)={\sum }_{n\ge 1}{h}_{n}J(a{)}^n,\mathrm{\forall }a\in B_A(0;r)\mathrm{.}$$If,inaddition,$H$iszeroproductpreservingonthewhole$B_A(0;r)$,then$J$ is an algebra isomorphism. %Similar conclusions hold for orthogonally additive $n$-homogeneous polynomials which are $n$-isometries.