The structure of invertible isometries of \(L^p[0,1]\), \(1\leq p<\infty\), \(p\neq 2\), were described byS. Banach in his 1932 classic [Théorie des opérations linéaires. Warszawa: Seminarium Matematyczne Uniwersytetu Warszawskiego; Warszawa: Instytut Matematyczny PAN (1932;Zbl 0005.20901;JFM 58.0420.01)]. This result was completely proved and generalised to \(L^p\) spaces on \(\sigma\)-finite measure spaces byJ. Lamperti [Pac. J. Math. 8, 459–466 (1958;Zbl 0085.09702)]. The paper under review studies the Banach algebra \(F^p(v,v^{-1})\) generated by an invertible isometry \(v\) of an \(L^p\) space (and its inverse). These are basic examples of \(L^p\)-operator algebras, i.e., Banach algebras that can be isometrically represented as operators on some \(L^p\)-space. \(F^p(G)\) and \(F^p_{\lambda}(G)\), the \(L^p\)-operator algebra and the reduced \(L^p\)-operator algebra of a locally compact group \(G\), are important examples. Here the examples \(F^p(\mathbb{Z})\) and \(F^p(\mathbb{Z}_n)\) are discussed in detail.
The concept of a ‘spectral configuration’ \(\sigma\) is defined and a Banach algebra \(F^p(\sigma)\) is associated to such a \(\sigma\). A strong dichotomy is obtained for these: They are isomorphic to either \(F^p(\mathbb{Z})\) or to the space of all continuous functions on their maximal ideal space. The structure of isometric isomorphisms between \(L^p\)-spaces, \(p\neq 2\), is discussed and it is shown that, under some assumptions, every isometric isomorphism between \(L^p\)-spaces is a combination of a multiplication operator and a nullset preserving bi-measurable isomorphism between the measure spaces. The main result on \(F^p(v,v^{-1})\) shows that, for a separable \(L^p\) space, the Gelfand transform defines an isometric isomorphism between \(F^p(v,v^{-1})\) and \(F^p(\sigma(v))\), where \(\sigma(v)\) is the spectral configuration associated to \(v\). A consequence is that \(F^p(v,v^{-1})\) can be represented on \(\ell^p\). Question raised: When is a separable \(L^p\)-operator algebra isometrically represented on \(\ell^p\)?
It is shown that \(F^p(v, v^{-1})\) is always semisimple; further, except for the case when it is isomorphic to \(F^p(\mathbb{Z})\), \(p\neq 2\), it is closed under the continuous functional calculus and the Gelfand transform is an isomorphism. Using the description of \(F^p(v, v^{-1})\) obtained, it is shown that there is a quotient of \(F^1(\mathbb Z)\) that cannot be isometrically represented on any \(L^p\)-space for any \(p\in [1,\infty)\). In particular, the class of Banach algebras that act on \(L^1\)-spaces is not closed under quotients, thus answering negatively, for \(p=1\), a question raised byC. Le Merdy [Math. Proc. Camb. Philos. Soc. 119, No. 1, 83–90 (1996;Zbl 0847.46029), Problem 3.8].

Reviewer: K. Parthasarathy (Chennai)

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.