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

Electrically controlled Bragg resonances of an ambichiral electro-optic structure: oblique incidence

The Pockels effect can increase the effective birefringence of ambichiral, electro--optic rejection filters made of materials with a $\bar{4}2m$ point group symmetry, when a dc electric field is applied parallel to the axis of nonhomogeneity. The reflectances and the transmittances of such an ambichiral structure for obliquely incident plane waves is solvable through a boundary-value problem that is formulated using the frequency-domain Maxwell equations, the constitutive equations that contain the Pockels effect, and standard algebraic techniques for handling 4x4 matrix ordinary differential equations. The Bragg resonance peaks, for different circular-polarized-incidence conditions, blueshift as the angle of incidence increases. These blueshifts are unaffected by the sign of the dc electric field.Comment: 11 page