Noncommmutative theorems: Gelfand Duality, Spectral, Invariant Subspace, and Pontryagin Duality

Details Noncommmutative theorems: Gelfand Duality, Spectral, Invariant Subspace, and Pontryagin Duality Noncommmutative theorems: Gelfand Duality, Spectral, Invariant Subspace, and Pontryagin Duality

2005-03-07

arxiv.org/abs/math/0503127v1

Mathematics

Operator Algebras

10 pages

Scientific paper

We extend the Gelfand-Naimark duality of commutative C*-algebras, "A COMMUTATIVE C*-ALGEBRA -- A LOCALLY COMPACT HAUSDORFF SPACE" to "A C*-ALGEBRA--A QUOTIENT OF A LOCALLY COMPACT HAUSDORFF SPACE". Thus, a C*-algebra is isomorphic to the convolution algebra of continuous regular Borel measures on the topological equivalence relation given by the above mentioned quotient. In commutative case this reduces to Gelfand-Naimark theorem. Applications: 1) A simultaneous extension, to arbitrary Hilbert space operators, of Jordan Canonical Form and Spectral Theorem of normal operators 2) A functional calculus for arbitrary operators. 3) Affirmative solution of Invariant Subspace Problem. 4) Extension of Pontryagin duality to nonabelian groups, and inevitably to groups whose underlying topological space is noncommutative.

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