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Harmonic analysis of weighted $$L^p$$-algebras. (English) Zbl 1247.43004
Throughout $$G$$ will always denote a compactly generated locally compact group. It is well known that $$L^1(G)$$ is a Banach algebra, where the multiplication is ordinary convolution of functions. For $$1 < p < \infty$$ it is also well known that $$L^p(G)$$ is not a convolution algebra. However, for certain weight functions $$\omega$$ on $$G, L^p(G, \omega)$$ is a Banach $$\ast$$-algebra. The aim of the paper under review is to extend the theory of convolution algebras to $$L^p(G, \omega)$$ when it is a Banach $$\ast$$-algebra. It is also assumed in this paper that $$G$$ has polynomial growth, in addition to the properties mentioned above. The authors give conditions on the weight $$\omega$$ that guarantees $$L^p(G, \omega)$$ is a Banach $$\ast$$-algebra. Furthermore, they say $$(G, \omega)$$ satisfies (LPAlg) if $$\omega$$ meets these particular conditions. Several examples of weights $$\omega$$ are given for which $$(G, \omega)$$ satisfies (LPAlg). An example of a weight $$\omega$$ is also given for which $$L^p(G, \omega)$$ is not an algebra. We shall say $$\omega$$ satisfies (BDna) if $$\omega$$ satisfies a Beurling-Domar type condition.
The authors proceed to give results involving symmetry, regularity, the weak Wiener property, and the Wiener property, when $$(G, \omega)$$ satisfies (LPAlg). An example of such a result is the following:
Let $$(G, \omega)$$ be (LPAlg). Let us assume that the weight $$\omega$$ also satisfies (BDna). Then the algebra $$L^p(G, \omega)$$ has the weak Wiener property.
Let $$p \geq 1$$ be a real number. In the last section of the paper a weight $$\omega$$ is constructed, depending on $$p$$, on $$\mathbb{R}$$ for which the commutative algebra $$L^p(G, \omega)$$ is symmetric. The authors then construct an infinite-dimensional topologically irreducible representation of $$L^p(G, \omega)$$. This is interesting since all topologically irreducible $$\ast$$-representations of $$L^p(G, \omega)$$ are one-dimensional.

##### MSC:
 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 22D15 Group algebras of locally compact groups 22D20 Representations of group algebras
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