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

Bhatt, S. J. (1998) Stinespring representability and Kadison's Schwarz inequality in non-unital Banach star algebras and applications Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 108 (3). pp. 283-303. ISSN 0253-4142

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Official URL: http://www.ias.ac.in/j_archive/mathsci/108/3/283-3...

Related URL: http://dx.doi.org/10.1007/BF02844483

Abstract

A completely positive operator valued linear map φon a (not necessarily unital) Banach *-algebra with continuous involution admits minimal Stinespring dilation iff for some scalar k < 0, φ(x)*φ(x)≤ kφ(x*x) for all x iff ∅ is hermitian and satisfies Kadison's Schwarz inequality ∅ (h) 2 k∅k ∅ (h 2) for all hermitianh iff φ extends as a completely positive map on the unitization Ae of 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 group G, 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.

Item Type:Article
Source:Copyright of this article belongs to Indian Academy of Sciences.
Keywords:Stinespring Representability; completely Positive Map; Kadison's Schwarz Inequality; Automatic Representability; Positive Definite Functions on a Group; Bochner Theorem
ID Code:59678
Deposited On:07 Sep 2011 05:16
Last Modified:18 May 2016 10:09

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