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 #Algebra over a field #Algebra representation #Bounded function #Combinatorics #Convexity #Discrete mathematics #Extreme point #FOS: Mathematics #Factorization #Functional Analysis (math.FA) #Hilbert space #Holomorphic and Operator Theory #Mathematical analysis #Mathematics #Nest algebra #Non-associative algebra #Operator Algebras (math.OA) #Pure mathematics #Separable space #Space (punctuation) #Topological space #Type (biology) #Weak topology (polar topology) #math.FA #math.OA #msc:46L30 #msc:46L55 #msc:47L35
paper · pdf · doi:10.48550/arxiv.2103.09600
published in arXiv (Cornell University) (Cornell University) · 26 pages; Example 3.9 and Remark 5.4 added. Some typos fixed. To appear in J. Funct. Anal
openalex publication_date 2021/03/17 · arxiv created 2022/01/14 · arxiv updated 2022/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.