Orthogonally additive holomorphic maps between C*-algebras

TL;DR

This paper characterizes orthogonally additive holomorphic maps between C*-algebras that preserve zero products, providing explicit forms involving symbol maps and multipliers, and establishing conditions for algebra isomorphisms.

Contribution

It offers a new structural description of zero product preserving orthogonally additive holomorphic maps between C*-algebras, including the commutative and general cases.

Findings

In the commutative case, maps are expressed via weight functions and a symbol map.

In the general case, conformal maps are characterized by central multipliers and Jordan isomorphisms.

If zero product preservation extends to the whole domain, the Jordan isomorphism becomes an algebra isomorphism.

Abstract

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 $\varphi: Y\to X$ such that $$ H(f)=\sum_{n\geq1} h_n (f\circ\varphi)^n, \quad\forall f\in B_{C_0(X)}(0;r). $$ In the general case, we show that if $H$ is also conformal then there exist central multipliers $h_n$'s of $B$ and a surjective Jordan isomorphism $J: A\to B$ such that $$ H(a) = \sum_{n\geq1} h_n J(a)^n, \quad\forall a\in B_A(0;r). $$ If, in addition, $H$ is zero product preserving on the whole $B_A(0;r)$, then $J$ is an algebra isomorphism. %Similar conclusions hold for orthogonally additive $n$-homogeneous polynomials which are $n$-isometries.