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

TL;DR

This paper presents a comprehensive method for decomposing representations of C*-algebras into irreducible components and applies it to solve three major problems in the field, advancing understanding of operator systems.

Contribution

It introduces a complete decomposition technique for states and representations of C*-algebras, enabling solutions to longstanding problems and conjectures in operator algebra theory.

Findings

Decomposition of states into pure states achieved

Representation decomposition into irreducible components demonstrated

Solutions provided for three major open problems in C*-algebras

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.