Operator Algebras II: Graph C*-Algebras

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Graph C*-algebras are a class of C*-algebras constructed from directed graphs. They form a bridge between algebra, analysis, and combinatorics. Their origins trace back to the 1980s with the work of Cuntz and Krieger, who introduced what are now called Cuntz–Krieger algebras. These arose in the study of Markov processes and symbolic dynamics.

In the 1990s, it was realized that Cuntz–Krieger algebras could be seen as special cases of a broader class: graph C*-algebras. Since then, a rich field of research has developed, touching on dynamical systems, K-theory, Morita equivalence, groupoid theory, and more.

Graph C*-algebras are not only concrete and accessible but also offer deep insights into structural questions: When is a C*-algebra simple? What does its ideal structure look like? How can such algebras be classified? These questions can often be tackled explicitly in the graph-theoretic setting.

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