Karen Strung
Graph $C^*$-algebras and their applications to quantum spaces
This short course will serve as an introduction to $C^*$-algebras through the lens of graph algebras, with an emphasis on their role as noncommutative topological spaces. After a short introduction to the basics of $C^*$-algebras, we will show how to construct of $C^*$-algebras from directed graphs. Graph $C^*$-algebras are a class of examples which are both rich and tractable. In particular, we will see how the combinatorics of the graph can determine key $C^*$-algebraic properties such as ideal structure and K-theory.
Building on this foundation, we will explore how certain $C^*$-algebras arising in noncommutative geometry, which we think of as algebras of functions on “quantum spaces”, can be realized within this framework, by considering the $C^*$-algebras of quantum flag manifolds in the sense of Drinfeld and Jimbo.
