Algebraic topology of $C^{\ast }$-algebras

Any $C^{\ast }$-algebra can be regarded as a generalization of locally compact, Hausdorff topological space $\mathcal{X}$. From the commutative commutative Gelfand-Naĭmark theorem it follows that the spectrum of any commutative $C^{\ast }$-algebra is a locally compact, Hausdorff space which have the exact information of the $C^{\ast }$-algebra. Here we consider a Gelfand spaces of $C^{\ast }$-algebras which can be regarded as a generalization of the spectrum. In case of commutative $C^{\ast }$-algebras the Gelfand space coincides with the spectrum. Generally Gelfand spaces are not Hausdorff and provide more detailed information of noncommutative $C^{\ast }$-algebras. Sometimes the Gelfand space contains the full information of noncommutative $C^{\ast }$-algebra. Usage of Gelfand spaces of $C^{\ast }$-algebra enables us to define some $C^{\ast }$-algebraic analogs of several notions of the classical algebraic topology.