Mackey imprimitivity and commuting tuples of homogeneous normal operators

Gadadhar Misra (), E. K. Narayanan () and Cherian Varughese ()
Additional contact information
Gadadhar Misra: Indian Statistical Institute
E. K. Narayanan: Indian Institute of Science
Cherian Varughese: Renaissance Communications

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 3, 1010-1025

Abstract: 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 $$\rho $$ ρ of $$C_0({\mathbb {S}})$$ C 0 ( S ) (where $${\mathbb {S}}$$ S is a locally compact Hausdorff space) into a direct sum. If there is a group G acting transitively on $${\mathbb {S}}$$ S and is adapted to the $$*$$ ∗ - representation $$\rho $$ 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 $$S\subset {\mathbb {C}}^d$$ S ⊂ C d 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.

Keywords: imprimitivity; induced representation; homogeneous operator; Primary 22D30; 22D45; 47B15 (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-024-00644-x Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:55:y:2024:i:3:d:10.1007_s13226-024-00644-x

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-024-00644-x

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().