Extensions of uniform algebras. (English) Zbl 1472.46050

Morley, Sam

Banach J. Math. Anal. 13, No. 4, 837-863 (2019).

Cole’s counterexample to the peak point conjecture [B. J. Cole, One-point parts and the peak point conjecture. Ph.D. dissertation, Yale University, New Haven, CT (1968)] provides a construction of extensions of uniform algebras having various desirable properties, and has been used in the theory of uniform algebras (e.g., [J. F. Feinstein, Stud. Math. 148, No. 1, 67–74 (2001; Zbl 1055.46035); Proc. Am. Math. Soc. 132, No. 8, 2389–2397 (2004; Zbl 1055.46036)]).
The paper under review introduces some new classes of uniform algebra extensions and investigates the properties that are inherited by any uniform algebra extensions such as being nontrivial, natural, regular and normal. The relationship between peak sets in the weak sense and the Shilov boundary of an extension to those of the original uniform algebra is studied as well.
Let \(X\) and \(Y\) be compact Hausdorff spaces, let \(A\) be a uniform algebra on \(X\) and \(B\) be a uniform algebra on \(Y\). If there exists a continuous surjection \(\Pi: Y \to X\) such that \(\Pi^*(A) \subseteq B\), then \(B\) is called a uniform algebra extension of \(A\). The author introduces a class of uniform algebra extensions called generalised Cole extensions, namely, if there exists a continuous surjection \(\Pi: Y \to X\) and a unital linear map \(T: C(X) \to C(Y)\) with \(\|T\|= 1\) such that \(T\circ \Pi^*= \mathrm{ id}_{C(X)}\), \(\Pi^*(A) \subseteq B\) and \(T(B)= A\). In this case, \(\Pi\) and \(T\) are called associated maps to the extension. It is shown in the paper, Section 5, that generalised Cole extensions preserve several properties of the original uniform algebra. The author also studies generalised Cole extensions implemented by a compact group. In general, a generalised Cole extension need not be implemented by a group; however, if the associated maps to the extension \(B\) satisfy \(\|\mathrm{id}_{C(Y)}- \Pi^*\circ T\|= 1\), then there exists a finite group \(G\) such that \(B\) is a generalised Cole extension implemented by \(G\). The paper is concluded with some examples and two open questions.

Reviewer: Ying-Fen Lin (Belfast)

MSC:

46J10	Banach algebras of continuous functions, function algebras
47B48	Linear operators on Banach algebras

Keywords:

uniform algebras; uniform algebra extensions; averaging operators; Cole extension

Citations:

Zbl 1055.46035; Zbl 1055.46036

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