Algebraic topology of $C^*$-algebras

Abstract

Any $C^*$-algebra can be regarded as a generalization of locally compact, Hausdorff topological space $\mathcal X$. From the commutative commutative Gelfand-Na\u{\i}mark theorem it follows that the spectrum of any commutative $C^*$-algebra is a locally compact, Hausdorff space which have the exact information of the $C^*$-algebra. Here we consider a Gelfand spaces of $C^*$-algebras which can be regarded as a generalization of the spectrum. In case of commutative $C^*$-algebras the Gelfand space coincides with the spectrum. Generally Gelfand spaces are not Hausdorff and provide more detailed information of noncommutative $C^*$-algebras. Usage of Gelfand spaces of $C^*$-algebra enables us to define some $C^*$-algebraic analogs of several notions of the classical algebraic topology, e.g. cohomology, fundamental group, multiplicative structure of $K$-theory and Adams operations