|
Abstract: C^*-algberas are a special class of norm-closed algerbas of operators on Hilbert space that have special connections with in topology, geometry, harmonic analysis, etc. Any COMMUTATIVE C*-algebra is naturally isomorphic to the algebra of complex functions on a compact Hausdorff space naturally associated with it. This "Gelfand-Naimark duality" sets up an equivalence between the category of commutative C*-algebras on one hand, and compact Hausdorff spaces on the other. This fact constitutes the heavenly gateway to practically everything we know about commutative C*-algebras. Furthermore, it leads to spectral theorem for normal operators on Hilbert space, Pontryagin duality for locally compact abelian groups, etc. We extend this (commutative) Gelfand-Naimark duality to the general (possibly noncommutative) C^*-algebras, by identifying the noncommutative analog of compact Hausdorff spaces. This analog turns out to be simply a quotient of a compact Hausdorff space. Again, this extends the equivalence of categories mentioned above to: (C*-algebras <----> Quotients of Compact Hausdorff spaces). We can only hope that this can open some heavenly doors to the study of general C*-algebras. We mention couple of applications--- |