This is an announcement for the paper "Noncommmutative Gelfand duality for not necessarily unital $C^*$-algebras, Jordan canonical form, and the existence of invariant subspaces" by Mukul S. Patel.

Abstract: Gelfand-Naimark duality (Commutative $C^*$-algebras $\equiv$ Locally compact Hausdorff spaces) is extended to \begin{center}$C^*$-algebras $\equiv$ Quotient maps on locally compact Hausdorff spaces.\end{center} 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.

Archive classification: Functional Analysis; Operator Algebras

Mathematics Subject Classification: 46L05; 47A13; 47A13; 43A40; 22B05

Remarks: Under consideration for publication by Electronic Reasearch

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http://arXiv.org/abs/math.FA/0508545

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