Mathematics

C*-algebras are operator algebras that form the conceptual foundation of noncommutative geometry. Since commutative C*-algebras yield categories anti-equivalent to categories of locally compact Hausdorff spaces by the celebrated Gelfand-Naimark equivalence theorem, noncommutative C*-algebras are viewed as function algebras on quantum spaces. Their study from this point of view is referred to as noncommutative topology. Here KK-theory and index theory are among prime tools leading to significant applications. Graph C*-algebras form a narrow but particularly inspiring class in which the noncommutative topology of quantum spaces is given by the combinatorics of oriented graphs. Graphs provide a very tangible grip on their C*-algebras allowing one to obtain deeper and more explicit results than are attainable in a general abstract setting. In particular, graph C*-algebras are formidably good in instantiating the general theory of C*-algebras. The aim of this lecture course is to introduce the basics of C*-algebras while enjoying educating fun with oriented graphs.