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

Qingying Bu¹ ; Ming-Hsiu Hsu² ; Ngai-Ching Wong³

¹ Department of Mathematics University of Mississippi University, MS 38677, U.S.A.
² Department of Mathematics National Central University Chung-Li, 32054, Taiwan
³ Department of Applied Mathematics National Sun Yat-sen University Kaohsiung, 80424, Taiwan

Studia Mathematica, Tome 234 (2016) no. 3, pp. 195-216

Résumé

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.

Détail

DOI : 10.4064/sm7922-6-2016

Keywords: * algebras ball centered radius orthogonally additive holomorphic map zero product preserving positive elements commutative there exist weight functions symbol map varphi sum geq circ varphi hskip forall general conformal there exist central multipliers surjective jordan isomorphism sum geq a hskip forall addition zero product preserving whole algebra isomorphism study orthogonally additive n homogeneous polynomials which n isometries