Unique disintegration of arbitrary positive definite functions on \(*\)- divisible semigroups
Let \((S,*)\) denote an Abelian semigroup with involution and denote by \(S^*\) the character space of \(S\), i.e. the set of \(*\)-homomorphisms into the semigroup \(\mathbb{C}\) with conjugation as involution. It is known [\textit{R. J. Lindahl} and \textit{P. H. Maserick}, Duke Math. J. 38, 771-782 (1971; Zbl 0243.43004)] that every bounded positive definite function \(\varphi\) on \(S\) is the `Laplace' transform \(\hat\mu: s\mapsto\int_ S\sigma(s)d\mu(s)\) of a unique measure \(\mu\) on \(\hat S\), the space of bounded characters of \(S\). \textit{C. Berg}, \textit{J. P. R. Christenson} and \textit{P. Ressel} [Harmonic Analysis on semigroups (Springer 1984; Zbl 0619.43001)] call \(S\) perfect if every positive definite function \(\varphi\) is the Laplace transform of a unique positive Radon measure on \(S^*\). However there is no such Laplace transform for a positive function on \(\mathbb{R}_ +\). To be able to achieve this the authors use the Kolmogorov construction for measures rather than restricting themselves to positive Radon measures. Let \(\mathcal A\) be the smallest \(\sigma\)-algebra on \(S^*\) induced by the evaluation mapping \(\sigma\mapsto\sigma(s): S^*\to\mathbb{C}\). The authors permit measures \(\mu\) on \((S^*,\mathcal A)\) such that \(\sigma\mapsto\sigma(s)\) is \(\mu\)-integrable for all \(s\in S\), calling \(S\) quasi-perfect if each positive definite function is the Laplace transform of a unique such measure. For \(S\) countable, Borel measures are Radon so perfection is equivalent to quasiperfection. The authors show that quasiperfection is preserved by \(*\)-homomorphic images and products, as for perfection, and that it is preserved by arbitrary direct sums (as opposed to the corresponding result for direct sums of sequences of perfect spaces). They define a \(*\)-semigroup to be \(*\)- divisible if every \(s\in S\) can be written in the form \(s=mt+nt^*\), \(t\in S\), \(m+n\geq 2\). (For example an Abelian group \(G\) with \(s^*=-s\) is \(*\)-divisible, though not divisible in the usual sense). They show that certain subsemigroups \(T\) of \(\mathbb{Q}^ 2\), with \((x,y)^*=(y,x)\), are perfect, and so also quasiperfect as they are countable. If \(S\) is a \(*\)-divisible Abelian semigroup, for each \(a\in S\) there is a subsemigroup \(T_ a\) as above and a \(*\)-homomorphism of \(\oplus T_ a\) onto \(S\). Hence the main theorem is that every \(*\)-divisible Abelian semigroup is quasi-perfect.
scientific article; zbMATH DE number 3694