Bochner's theorem for finite inverse semigroups and its connection to Choi's theorem
Sohail, Sahil
Abstract
Bochner's theorem characterizes positive definite functions on groups through the positivity of their Fourier transforms and plays a fundamental role in Harmonic analysis. While Bochner-type results are known for certain classes of semigroups, they typically differ from the group theoretic formulations and do not retain the same level of simplicity and generality. In this work, we prove a Bochner-type theorem for finite inverse semigroups at the level of matrix valued linear maps on the contracted algebras of the semigroups. Using the intrinsic partial order of inverse semigroups, positivity naturally arises through a Möbius-transformed map. Our main result characterizes the positive definiteness of the Möbius transformed map in terms of the positivity of the Fourier transform of the original map with respect to a complete family of inequivalent irreducible representations of the contracted algebra induced by irreducible unitary representations of the maximal subgroups of the inverse semigroup. The proof relies on Fourier inversion formula, Schur orthogonality relations, and alternative characterizations of positive definite maps, all established here in the setting of finite inverse semigroups. As a special case, we show that for the inverse semigroup of matrix units, Bochner's theorem reduces exactly to Choi's characterization of completely positive maps.