Weak multiplicative operators on function algebras without units

Volume 91 / 2010

Thomas Tonev Banach Center Publications 91 (2010), 411-421 MSC: Primary 46J10, 46J20; Secondary 46H40. DOI: 10.4064/bc91-0-24

Abstract

For a function algebra $A$ let $\partial A$ be the Shilov boundary, $\delta A$ the Choquet boundary, $p(A)$ the set of $p$-points, and $|A|=\{|f|\colon f\in A\}$. Let $X$ and $Y$ be locally compact Hausdorff spaces and $A\subset C(X)$ and $B\subset C(Y)$ be dense subalgebras of function algebras without units, such that $X=\partial A$, $Y=\partial B$ and $p(A)=\delta A$, $p(B)=\delta B$. We show that if $ \Phi\colon |A|\to|B|$ is an increasing bijection which is sup-norm-multiplicative, i.e. $\|\Phi(|f|)\,\Phi(|g|)\|=\|fg\|$, $f,g\in A$, then there is a homeomorphism $\psi\colon p(B)\to p(A)$ with respect to which $\Phi$ is a $\psi$-composition operator on $p(B)$, i.e. $(\Phi(|f|))(y)=|f(\psi(y))|$, $f\in A,\ y\in p(B)$. We show also that if $A\subset C(X)$ and $B\subset C(Y)$ are dense subalgebras of function algebras without units, such that $X=\partial A$, $Y=\partial B$ and $p(A)=\delta A$, $p(B)=\delta B$, and $ T\colon A\to B$ is a sup-norm-multiplicative surjection, namely, $\|Tf\, Tg\|=\|fg\|$, $f,g\in A$, then $T$ is a $\psi$-composition operator in modulus on $p(B)$ for a homeomorphism $\psi\colon p(B)\to p(A)$, i.e. $|(Tf)(y)|=|f(\psi(y))|$, $f\in A$, $y\in p(B)$. In particular, $T$ is multiplicative in modulus on $p(B)$, i.e. $|T(fg)|=|Tf\,Tg|$, $f,g\in A$. We prove also that if $A\subset C(X)$ is a dense subalgebra of a function algebra without unit, such that $X=\partial A$ and $p(A)=\delta A$, and if $ T\colon A\to B$ is a weakly peripherally-multiplicative surjection onto a function algebra $B$ without unit, i.e. $\sigma_\pi(Tf\, Tg)\cap\sigma_\pi(fg)\neq\emptyset$, $f,g\in A$, and preserves the peripheral spectra of algebra elements, i.e. $\sigma_\pi(Tf)=\sigma_\pi(f)$, $f\in A$, then $T$ is a bijective $\psi$-composition operator on $p(B)$, i.e. $(Tf)(y)=f(\psi(y))$, $f\in A$, $y\in p(B)$, for a homeomorphism $\psi\colon p(B)\to p(A)$. In this case $A$ is necessarily a function algebra and $T$ is an algebra isomorphism. As a consequence, a multiplicative operator $T$ from a dense subalgebra $A\subset C(X)$ of a function algebra $B$ without unit, such that $X=\partial A$ and $p(A)=\delta A$, onto a function algebra without unit $B$ is a sup-norm isometric algebra isomorphism if and only if $T$ is weakly peripherally-multiplicative and preserves the peripheral spectra of algebra elements. The results extend to function algebras without units a series of previous results for algebra isomorphisms.

Authors

  • Thomas TonevThe University of Montana, Missoula
    Missoula, MT 59812, USA
    e-mail

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