Banach Journal of Mathematical Analysis

Banach function algebras and certain polynomially norm-preserving maps

Maliheh Hosseini and Fereshteh Sady

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Abstract

Let $A$ and $B$ be Banach function algebras on compact Hausdorff spaces $X$ and $Y$, respectively. Given a non-zero scalar $\alpha$and $s,t\in \Bbb N$ we characterize the general form of suitable powers of surjective maps $T, T': A \longrightarrow B$ satisfying $\|(Tf)^s (T'g)^t-\alpha\|_Y=\|f^s g^t-\alpha \|_X$, for all $f,g \in A$, where $\|\cdot \|_X$ and $\|\cdot \|_Y$ denote the supremum norms on $X$ and $Y$, respectively. A similar result is given for the case where $T=T'$ and $T$ is defined between certain subsets of $A$ and $B$. We also show that if $T: A\longrightarrow B$ is a surjective map satisfying the stronger condition$R_\pi((Tf)^{s}(Tg)^{t}-\alpha)\cap R_\pi(f^{s}g^{t}-\alpha)\neq\varnothing $ for all $f,g \in A$, where $R_\pi(\cdot)$ denotes the peripheral range of the algebra elements, then there exists a homeomorphism $\varphi$ from the Choquet boundary $c(B)$ of $B$ onto the Choquet boundary $c(A)$ of $A$ such that $(Tf)^{d}(y)=(T1)^{d}(y)\,(f \circ \varphi(y))^{d}$ for all $f\in A$ and $y\in c(B)$,where $d$ is the greatest common divisor of $s$ and $t$.

Article information

Source
Banach J. Math. Anal., Volume 6, Number 2 (2012), 1-18.

Dates
First available in Project Euclid: 13 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1342210157

Digital Object Identifier
doi:10.15352/bjma/1342210157

Mathematical Reviews number (MathSciNet)
MR2945985

Zentralblatt MATH identifier
1256.46030

Subjects
Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 47B48: Operators on Banach algebras 46J20: Ideals, maximal ideals, boundaries

Keywords
Banach function algebra polynomially norm-preserving map peripheral spectrum peripheral range Choquet boundary

Citation

Hosseini , Maliheh; Sady , Fereshteh. Banach function algebras and certain polynomially norm-preserving maps. Banach J. Math. Anal. 6 (2012), no. 2, 1--18. doi:10.15352/bjma/1342210157. https://projecteuclid.org/euclid.bjma/1342210157


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