A USER'S GUIDE TO THE CLASSIFICATION OF GRAPH C*-ALGEBRAS

Graph C*-algebras (and their precursors, the Cuntz-Krieger algebras) are ubiquitous in modern C*-algebra theory and pop up regularly in noncommutative geometry and/or as models for quantum groups and spaces. In recent years, there has been significant progress concerning the classification of graph C*-algebras by K-theoretical invariants, not just addressing stable isomorphisms as in the early days of the theory, but also allowing for analysis of when two such C*-algebras are isomorphic in a way preserving naturally occurring structures such as diagonals and circle actions. The results are particularly strong for unital graph C*-algebras, where they allow completely general ideal lattices and, in many cases, come paired with results that show that deciding *-isomorphism may, at least in principle, be left to a computer. It is my ambition to give a complete overview of what we know - and do not know - about questions of this nature for the benefit of the noncommutative geometer and/or quantum-group expert who happens to encounter two or more of such C*-algebras and wonders whether or not they are the same. The strongest results available at this time draw extensively on discrete methods that argue directly on graphs or their matrix representations. As these pictures are sometimes superior for computations, I will give a good deal of background on them. This talk is based on my joint work with various subsets of {Arklint, Armstrong, Brix, Carlsen, Dor-On, Gabe, Katsura, Ortega, Restorff, Ruiz, Sims, Sørensen, Tomforde}.