Wojciech Szymanski

Cuntz-Krieger algebras of infinite graphs and matrices

We show that the Cuntz-Krieger algebras of infinite graphs and infinite {0,1}-matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their K-theory. Since the finite approximating graphs have sinks, we have to calculate the K-theory of Cuntz-Krieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.

Quantum real projective space, disc and sphere

We define the C*-algebra of quantum real projective space RPq2, classify its irreducible representations, and compute its K-theory. We also show that the q-disc of Klimek and Lesniewski can be obtained as a non-Galois Z2-quotient of the equator Podleś quantum sphere. On the way, we provide the Cartesian coordinates for all Podleś quantum spheres and determine an explicit form of isomorphisms between the C*-algebras of the equilateral spheres and the C*-algebra of the equator one.

On semiprojectivity of C^*-algebras of directed graphs

It is shown that if E is a countable, transitive directed graph with finitely many vertices, then C * (E) is semiprojective.

Labeled trees and localized automorphisms of the Cuntz algebras

We initiate a detailed and systematic study of automorphisms of the Cuntz algebras On which preserve both the diagonal and the core UHF-subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of Aut(On) and leads to numerous new examples. In particular, we completely classify all such automorphisms of O2 for the permutation unitaries in ⊗ 4 M2. We show that the subgroup of Out(O2) generated by these automorphisms contains a copy of the infinite dihedral group Z ⋊ Z2. MSC 2000: 46L40, 46L05, 37B10

More localized automorphisms of the Cuntz algebras

We completely determine the localized automorphisms of the Cuntz algebras

Bimodules for Cuntz-Krieger algebras of infinite matrices

O_n

Computing the Jones index of quadratic permutation endomorphisms of O2

corresponding to permutation matrices in

Infinite simple C*-algebras and Reduced cross products of abelian C*-algebras and free groups

M_n \otimes M_n

The restricted Weyl group of the Cuntz algebra and shift endomorphisms

for

Noncommutative balls and mirror quantum spheres

n=3