  Gelfand duality between unital commutative C*-algebras and Compact Hausdorff
spaces is extended to all unital C*-algebras, where the dual objects are what
we call compact Hausdorff quantum spaces. We apply this result to obtain, a
characterization of unitary groups of C*-algebras, and, for arbitrary bounded
Hilbert space operators, (i) A spectral theorem cum continuous functional
calculus, and (ii) A proof of the general Invariant Subspace Theorem. Also
described is a nonabelian generalization of Pontryagin duality of abelian
locally compact groups.
