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

Tom 234 / 2016

Qingying Bu, Ming-Hsiu Hsu, Ngai-Ching Wong Studia Mathematica 234 (2016), 195-216 MSC: 17C65, 46G25, 46L05, 47B33. DOI: 10.4064/sm7922-6-2016 Opublikowany online: 26 August 2016


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.


  • Qingying BuDepartment of Mathematics
    University of Mississippi
    University, MS 38677, U.S.A.
  • Ming-Hsiu HsuDepartment of Mathematics
    National Central University
    Chung-Li, 32054, Taiwan
  • Ngai-Ching WongDepartment of Applied Mathematics
    National Sun Yat-sen University
    Kaohsiung, 80424, Taiwan

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