A Complete Solution to the Problem of Decomposing a Representation Into Irreducible Representations and its Applications to the Solutions of Three Great Problems in C*-Algebras

2012/12/26 by Ansari, Shamim I
#46L05 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1212.6192

Abstract

In this paper we give a decomposition of a state on a C^*-algebra into a family of pure states and a decomposition of a representation into a family of irreducible representation. Then, we use it to solve the following three problems and/or conjectures.. (1) The noncommutative Stone-Weierstrass problem, (2) The extension problem (asked by Arveson) of a pure state on a nonseparable operator system to a boundary state on the generated C^*-algebra, and (3) The hyperrigidity problem of an operator system under the hypothesis that pure states have the unique extension property, conjectured by Arveson.

Related

• Rigidity of operator systems: tight extensions and noncommutative measurable structures
• A new obstruction to Arveson's hyperrigidity conjecture
• Arveson's hyperrigidity conjecture is false
• Restrictions of pure states to subspaces of C^*-algebras
• Unperforated pairs of operator spaces and hyperrigidity of operator systems