Noncommutative Balls and Mirror Quantum Spheres

Noncommutative analogues of n-dimensional balls are defined by repeated application of the quantum double suspension to the classical low-dimensional spaces. In the `even-dimensional' case they correspond to the Twisted Canonical Commutation Relations of Pusz and Woronowicz. Then quantum spheres are constructed as double manifolds of noncommutative balls. Both C*-algebras and polynomial algebras of the objects in question are defined and analyzed, and their relations with previously known examples are presented. Our construction generalizes that of Hajac, Matthes and Szymanski for `dimension 2', and leads to a new class of quantum spheres (already on the C*-algebra level) in all `even-dimensions'.Comment: 20 page

On conjugacy of maximal abelian subalgebras and the outer automorphism group of the Cuntz algebra

We investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of maximal abelian subalgebras in On. In particular, we exhibit an uncountable family of maximal abelian subalgebras, conjugate to the standard maximal abelian subalgebra Dn via Bogolubov automorphisms, that are not inner conjugate to Dn

Endomorphisms of the Cuntz Algebras and the Thompson Groups

We investigate the relationship between endomorphisms of the Cuntz algebra ${\mathcal O}_2$ and endomorphisms of the Thompson groups $F$, $T$ and $V$ represented inside the unitary group of ${\mathcal O}_2$. For an endomorphism $\lambda_u$ of ${\mathcal O}_2$, we show that $\lambda_u(V)\subseteq V$ if and only if $u\in V$. If $\lambda_u$ is an automorphism of ${\mathcal O}_2$ then $u\in V$ is equivalent to $\lambda_u(F)\subseteq V$. Our investigations are facilitated by introduction of the concept of modestly scaling endomorphism of ${\mathcal O}_n$, whose properties and examples are investigated.Comment: v1: 10 pages. v2: minor changes, updated reference list, 11 pages; to appear in Studia Mathematic

Stable rank of graph algebras. Type I graph algebras and their limits

For an arbitrary countable directed graph E we show that the only possible values of the stable rank of the associated Cuntz-Krieger algebra C*(E) are 1, 2 or \infty. Explicit criteria for each of these three cases are given. We characterize graph algebras of type I, and graph algebras which are inductive limits of C*-algebras of type I. We also show that a gauge-invariant ideal of a graph algebra is itself isomorphic to a graph algebra.Comment: 13 pages, LaTe