The algebra \(L^p(I,X)\) with order convolution and its multipliers (Q2468913)

This paper deals with harmonic analysis relating to an ordered semigroup by means of the construction of suitable measure convolution algebras and of \(L^{p}\) type spaces. Complications for semigroups arise as there is generally no Haar measure. The construction here introduces an extraneous element, denoted \(X\), so circumventing the real problem and dealing instead with the construction of \(L^{p}\) spaces with order convolution. The approach differs from that of \textit{E. Hewitt} and \textit{H. S. Zuckerman} for \(\ell^{1}\) [Trans. Am. Math. Soc. 83, 70--97 (1956; Zbl 0072.12701)] going through till \textit{M. Lashkarizadeh Bami} for \(L^{1}\) [Semigroup Forum 52, 389--392 (1996; Zbl 0854.43002)]. Let \(I\) denote an interval in \(\mathbf{R}\) with end-points \(a < b\), \((\Omega, \Sigma, \mu)\) a finite measure space, \(\mu\) being a Borel measure satisfying a suitable Radon-Nikodým property for \(\mu\) with respect to the Lebesgue measure on I. The author restricts herself to that \(I = (0,1]\). A basic \(L^{1}(I)\) theory for an interval \(I \in \mathbb R\) considered as a locally compact semigroup, given the product \(x.y = \max(x,y)\), was developed by \textit{L. J. Lardy} [Stud. Math. 27, 1--8 (1966; Zbl 0166.11701)] and the author's results extend those of Lardy. The Radon-Nikodým property is used to identify \(L^{1}(I)\) with a closed subalgebra of a convolution algebra \(M(I)\) of regular Borel measures on \(I\). Lardy develops a Gelfand representation identifying the maximal ideal space with the space of semi-characters and shows that the maximal ideal space \(\Delta (L^{1}(I))\) can be identified topologically with \((a,b]\). There is a Bochner-type theorem for positive definite functions, approximate units exist and there is a characterisation of multipliers. Similar theories are available for \(L^{p}\), \(1 \leq p \leq \infty\). \textit{J. W. Baker, J. S. Pym} and \textit{H. L. Vasudeva} [Mathematika 29, 42--54 (1982; Zbl 0477.46031)] constructed a commutative semigroup structure, compact in the order topology, based on \(I\). The author points out that the proof by Baker et al. on the existence of multipliers from \(L^p\) to \(L^r\) is incorrect unless \(p = r\). The author, instead of constructing a topological semigroup structure from \(I\), uses a unital commutative Banach algebra \(X\) which supports the measure \(\mu\). She defines \(L^{p}(I,X)\) spaces in such a way that they are semisimple Banach algebras with convolution and are contained in a measure-convolution algebra \(M(I,X)\), viz., the set of all \(\sigma\)-additive regular \(X\)-valued measures on the Borel sets of \(I\). The Radon-Nikodým derivatives of the elements of \(L^{p}(I,X)\) can be embedded in \(M(I,X)\) and the embedding preserves the semigroup structure. Providing \(M\) with a total-variation norm gives it a Banach space structure. The maximal ideal space \(\Delta L^{p}(I,X)\) is shown to be homeomorphic to \(I \times \Delta(X)\). The actual Gelfand transform is too abstractly characterised to be clear.