Orthogonally additive holomorphic maps between C*-algebras
Orthogonally additive holomorphic maps between C*-algebras
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DOI：
10.4064/sm7922-6-2016
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发表时间：
2015-12
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影响因子：
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通讯作者：
Q. Bu;Ming-Hsiu Hsu;N. Wong
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作者：
Q. Bu;Ming-Hsiu Hsu;N. Wong
设$A,B$为C*代数，$B_A(0;r)$为$A$中以$0$为中心半径为$r>0$的开放球，$H:B_A(0;r)\to B$为正交加性全纯映射。如果$H$在$B_A(0;r)$的正元素上是零积保持，我们证明，在$A=C_0(X)$和$B=C_0(Y)$的交换情况下，存在权函数$h_n$和符号映射$\varphi: Y\to X$，使得$$ H(f)=\sum_{n\geq1} h_n (f\circ\varphi)^n, \quad\forall f\in B_{C_0(X)}(0;r). $$在一般情况下，我们证明了如果$H$也是共形的，则存在$B$的中心乘子$h_n$ 's和一个满射Jordan同构$J: A\to B$，使得$$ H(a) = \sum_{n\geq1} h_n J(a)^n, \quad\forall a\in B_A(0;r). $$另外，如果$H$在整体上是零积保持$B_A(0;r)$，则$J$是一个代数同构。 %Similar conclusions hold for orthogonally additive $n$-homogeneous polynomials which are $n$-isometries.
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.