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Noncommmutative Gelfand Duality for not necessarily unital $C^*$-algebras, Jordan Canonical form, and the existence of invariant subspaces

Mukul S. Patel
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Abstract

Gelfand-Naimark duality (Commutative $C^*$-algebras $\equiv$ Locally compact Hausdorff spaces) is extended to $C^*$-algebras $\equiv$ Quotient maps on locally compact Hausdorff spaces. Using this duality, we give for an \emph{arbitrary} bounded operator on a complex Hilbert space of several dimensions, a functional calculus and the existence theorem for nontrivial invariant subspace.

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Mukul S. Patel. 2005-08-27. Noncommmutative Gelfand Duality for not necessarily unital $C^*$-algebras, Jordan Canonical form, and the existence of invariant subspaces. https://arxiv.org/abs/math/0508545

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