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

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 A_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 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.