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

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

2008-11-12

arxiv.org/abs/0811.1824v1

Mathematics

Operator Algebras

Scientific paper

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.

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