Operator algebra seminar

Speaker:  Sara Arklint (KU)

Title: A dynamical characterization of diagonal preserving isomorphisms of Cuntz-Krieger algebras

Abstract: In the theory of C*-algebras, one of the key families of illustrating and inspiring examples are those introduced by Cuntz and Krieger in 1980 as arising from topological Markov shifts.  They showed that flow equivalent shift spaces give stably isomorphic Cuntz-Krieger algebras, but Rørdam showed 15 years later that the converse does not hold in general.  Ever since, it has been an intrinsic question what the relationship between the dynamical system and its C*-algebra then is, and within the last few years, there have been rapid developments.

For simple Cuntz-Krieger algebras, this was beautifully answered by Matsumoto just 3 years ago, where he established that his notion of continuous orbit equivalence for shift spaces corresponds to an isomorphism between the Cuntz-Krieger algebras that preserve their diagonal, a canonical commutative subalgebra.  One year later, Matsumoto and Matui established a similar characterization of flow equivalence, still for simple Cuntz-Krieger algebras.  Already that same year, Brownlowe, Carlsen, and Whittaker extended Matsumoto's characterization to all Cuntz-Krieger algebras with Cuntz's property (I), and this year Eilers, Ruiz, and myself managed to remove this last restriction on the validity of the characterization.  I will describe these developments, as well as explain how we can now give a characterization of flow equivalence for Cuntz-Krieger algebras in full generality.