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We provide a complete description of those Banach algebras that are generatedby an invertible isometry of an $L^p$-space together with its inverse. Examplesinclude the algebra $PF_p(\mathbb{Z})$ of $p$-pseudofunctions on $\mathbb{Z}$,the commutative $C^*$-algebra $C(S^1)$ and all of its quotients, as well asuncountably many `exotic' Banach algebras.  We associate to each isometry of an $L^p$-space, a spectral invariant called`spectral configuration', which contains considerably more information than itsspectrum as an operator. It is shown that the spectral configuration describesthe isometric isomorphism type of the Banach algebra that the isometrygenerates together with its inverse.  It follows from our analysis that these algebras are semisimple. With theexception of $PF_p(\mathbb{Z})$, they are all closed under continuousfunctional calculus, and their Gelfand transform is an isomorphism.  As an application of our results, we show that Banach algebras that act on$L^1$-spaces are not closed under quotients. This answers the case $p=1$ of aquestion asked by Le Merdy 20 years ago.

Banach algebras generated by an invertible isometry of an $L^p$-space