Mackey imprimitivity and commuting tuples of homogeneous normal operators

Article Type

Research Article

Publication Title

Indian Journal of Pure and Applied Mathematics

Abstract

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting d- tuples of homogeneous normal operators. The Hahn–Hellinger theorem gives a canonical decomposition of a ∗- algebra representation ρ of C0(S) (where S is a locally compact Hausdorff space) into a direct sum. If there is a group G acting transitively on S and is adapted to the ∗- representation ρ via a unitary representation U of the group G, in other words, if there is an imprimitivity, then the Hahn–Hellinger decomposition reduces to just one component, and the group representation U becomes an induced representation, which is Mackey’s imprimitivity theorem. We consider the case where a compact topological space S⊂Cd decomposes into finitely many G- orbits. In such cases, the imprimitivity based on S admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of G- orbits.

First Page

1010

Last Page

1025

DOI

10.1007/s13226-024-00644-x

Publication Date

9-1-2024

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