LEAVITT PATH ALGEBRAS AND THE ALGEBRAIC KIRCHBERG-PHILLIPS PROBLEM

The Kirchberg-Phillips theorem says that unital separable nuclear purely infinite simple C*-algebras in the UCT class are classified by their (topological, C*-algebraic) K-theory and, more generally, that any two separable nuclear purely infinite simple C*-algebras that are KK-isomorphic are in fact isomorphic. This result was preceded by a similar result for Cuntz-Krieger C*-algebras (all of which are separable and nuclear) due to Cuntz and Rørdam. It is a long-standing open problem to determine whether a similar result holds for some sizable class of plain purely infinite simple algebras, including those appearing as Leavitt path algebras. The Kirchberg-Phillips theorem has two main ingredients: Kirchberg's Geneva theorems concerning tensor products with the Cuntz algebras O_2 and O_∞, both of which are known to fail in the purely algebraic case, and the Phillips theorem that describes KK-groups between any two C*-algebras in the KP class as the homotopy classes of homomorphisms, and also as asymptotic unitary equivalence classes of asymptotic homomorphisms. In this talk, I will discuss recent results which provide a description of algebraic and hermitian bivariant K-theory between purely infinite simple Leavitt path algebras in terms of homotopy classes and in terms of generalized conjugacy classes of homomorphisms, which we use to show that the Bowen-Franks group classifies such algebras up to involution preserving homotopy equivalence. Next, recall that the Cuntz-Rørdam theorem uses also the fact that a graph move called the Cuntz splice preserves the isomorphism class of purely infinite simple Cuntz-Krieger algebras. In particular, the Cuntz algebras O_2 and its splice O_{2^-} are isomorphic. It is not known whether the corresponding Leavitt path algebras L_2 and L_{2^-} are isomorphic, but existing results around this problem point in the negative direction. I shall also comment on further results in the same direction.