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

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.Comment: 10 page

Theory of Multiband Superconductivity in Iron Pnictides

The precise nature of unconventional superconductivity in Iron Pnictides is presently a hotly debated issue. Here, using insights from normal state electronic structure and symmetry arguments, we show how an unconventional SC emerges from the bad metal "normal" state. Short-ranged, multi-band spin- and charge correlations generates nodeless SC in the active planar $d_{xz,yz}$ bands, and an inter-band proximity effect induces out-of-plane gap nodes in the passive $d_{3z^{2}-r^{2}}$ band. While very good quantitative agreement with various key observations in the SC state and reconciliation with NMR and penetration depth data in the same picture are particularly attractive features of our proposal, clinching evidence would be an experimental confirmation of c-axis nodes in future work.Comment: 4 pages, 2 eps figures, submitted to PRL, text modifie