C-extreme maps and nests

Article Type

Research Article

Publication Title

Journal of Functional Analysis


The generalized state space SH(A) of all unital completely positive (UCP) maps on a unital C⁎-algebra A taking values in the algebra B(H) of all bounded operators on a Hilbert space H, is a C⁎-convex set. In this paper, we establish a connection between C⁎-extreme points of SH(A) and a factorization property of certain algebras associated to the UCP maps. In particular, the factorization property of some nest algebras is used to give a complete characterization of those C⁎-extreme maps which are direct sums of pure UCP maps. This significantly extends a result of Farenick and Zhou (1998) [14] from finite to infinite dimensional Hilbert spaces. Also it is shown that normal C⁎-extreme maps on type I factors are direct sums of normal pure UCP maps if and only if an associated algebra is reflexive. Further, a Krein-Milman type theorem is established for C⁎-convexity of the set SH(A) equipped with bounded weak topology, whenever A is a separable C⁎-algebra or it is a type I factor. As an application, we provide a new proof of a classical factorization result on operator-valued Hardy algebras.



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