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

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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 > 0, ϕ(x)*ϕ(x) ≤ kϕ(x*x) for allx iff ϕ is hermitian and satisfies Kadison’s Schwarz inequality ϕ(h) 2 ≤ kϕ(h 2) for all hermitianh iff ϕ extends as a completely positive map on the unitizationA e 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 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.

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

    S. J. Bhatt

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  1. S. J. Bhatt
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Bhatt, S.J. Stinespring representability and Kadison’s Schwarz inequality in non-unital Banach star algebras and applications. Proc. Indian Acad. Sci. (Math. Sci.) 108, 283–303 (1998). https://doi.org/10.1007/BF02844483

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  • DOI: https://doi.org/10.1007/BF02844483

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