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 compactHausdorff spaces by the celebrated Gelfand-Naimark equivalence theorem, noncommutative C*-algebrasare viewed as function algebras on quantum spaces. Their study from this point of view is referred toas noncommutative topology. Here KK-theory and index theory are among prime tools leading tosignificant applications. Graph C*-algebras form a narrow but particularly inspiring class in whichthe 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 moreexplicit results than are attainable in a general abstract setting. In particular, graph C*-algebras areformidably good in instantiating the general theory of C*-algebras. The aim of this lecture courseis to introduce the basics of C*-algebras while enjoying educating fun with oriented graphs..