I will focus on explaining why the study of C*-algebras is called non-commutative topology emphasizing the interdisciplinary character of the subject; the disciplines being algebra, analysis and topology.
The talk will feature exciting stuff like groups, Hilbert spaces (infinite-dimensional, of course), ideals in algebras/rings and topological spaces. Altogether redundant notions such as smoothness, fields of strictly positive characteristic, unbounded operators, physics and reality will be ignored.
The talk should be accessible to anyone with a basic knowledge of the exciting stuff mentioned above.