• Stinespring representability and Kadison’s Schwarz inequality in non-unital Banach star algebras and applications

• Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/108/03/0283-0303

• Keywords

Stinespring representability; completely positive map; Kadison’s Schwarz inequality; automatic representability; positive definite functions on a group; Bochner theorem

• Abstract

A completely positive operator valued linear map ϕ on a (not necessarily unital) Banach *-algebra with continuous involution admits minimal Stinespring dilation iff for some scalark &gt; 0, ϕ(x)*ϕ(x) ≤ kϕ(x*x) for allx iff ϕ is hermitian and satisfies Kadison’s Schwarz inequality ϕ(h)2≤ kϕ(h2) for all hermitianh iff ϕ extends as a completely positive map on the unitizationAeof A. A similar result holds for positive linear maps. These provide operator state analogues of the corresponding well-known results for representable positive functionals. Further, they are used to discuss (a) automatic Stinespring representability in Banach *-algebras, (b) operator valued analogue of Bochner-Weil-Raikov integral representation theorem, (c) operator valued analogue of the classical Bochner theorem in locally compact abelian groupG, and (d) extendability of completely positive maps from *-subalgebras. Evans’ result on Stinespring respresentability in the presence of bounded approximate identity (BAI) is deduced. A number of examples of Banach *-algebras without BAI are discussed to illustrate above results.

• Author Affiliations

1. Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar - 388 120, India

• Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019