2021/03/17 by B. V. Rajarama Bhat, Manish Kumar, Bhat, B. V. Rajarama +1
Mathematics · #46L30 #46L55 #47L35 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2103.09600
openalex publication_date 2021/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 map. In particular, this 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 [Proc. Amer. Math. Soc. 126 (1998)] 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.