Noncommutative Gelfand Duality and Applications I: The Existence of invariant subspaces

TL;DR

This paper extends Gelfand duality to all unital C*-algebras, introduces compact Hausdorff quantum spaces, and applies these concepts to characterize unitary groups, develop spectral theory, and prove the invariant subspace theorem.

Contribution

It introduces the concept of compact Hausdorff quantum spaces as duals of all unital C*-algebras and applies this to spectral theory and invariant subspaces.

Findings

Characterization of unitary groups of C*-algebras

Spectral theorem with continuous functional calculus for bounded operators

Proof of the general invariant subspace theorem

Abstract

Gelfand duality between unital commutative C*-algebras and Compact Hausdorff spaces is extended to all unital C*-algebras, where the dual objects are what we call compact Hausdorff quantum spaces. We apply this result to obtain, a characterization of unitary groups of C*-algebras, and, for arbitrary bounded Hilbert space operators, (i) A spectral theorem cum continuous functional calculus, and (ii) A proof of the general Invariant Subspace Theorem. Also described is a nonabelian generalization of Pontryagin duality of abelian locally compact groups.