We apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph C * -algebras. First we give a new proof of non-inner conjugacy of the diagonal MASA D E to its non-trivial image under a quasi-free automorphism, where E is a finite transitive graph. Then we exhibit a large class of MASAs in the Cuntz algebra O n that are not inner conjugate to the diagonal D n .

We investigate quantum lens spaces, C(L 2n+1 q (r; m)), introduced by Brzeziński-Szymański as graph C * -algebras. We give a new description of C(L 2n+1 q (r; m)) as graph C * -algebras amending an error in the original paper by Brzeziński-Szymański. Furthermore, for n ≤ 3, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r. This builds upon the work of Eilers, Restorff, Ruiz and Sørensen.

The problem of inner vs outer conjugacy of subalgebras of certain graph C * -algebras is investigated. For a large class of finite graphs E, we show that whenever α is a vertex-fixing quasi-free automorphism of the corresponding graph C * -algebraThat is, the two MASAs D E and α(D E ) of C * (E) are outer but not inner conjugate. For the Cuntz algebras O n , we find a criterion which guarantees that a polynomial automorphism moves the canonical UHF subalgebra to a non-inner conjugate UHF subalgebra. The criterion is phrased in terms of rescaling of trace on diagonal projections.

We study the noncommutative topology of the C * -algebras C(CP n q ) of the quantum projective spaces within the framework of Kasparov's bivariant K-theory. In particular, we construct an explicit KK-equivalence with the commutative algebra C n+1 . Our construction relies on showing that the extension of C * -algebras relating two quantum projective spaces of successive dimensions admits a splitting, which we can describe explicitly using graph algebra techniques.

We give a candidate of a vector space basis for the algebra O(S 4n−1 q ) of the quantum symplectic sphere for every n ≥ 1. The construction follows by a nontrivial application of the Diamond Lemma. The conjecture is supported by computer experiments for n = 1, 2, ..., 8. The work is motivated by a result of Landi and D'Andrea, who proved that the first n − 1 generators of the C * -algebra C(S 4n−1 q ), n ≥ 2 are zero. By finding a vector space basis, we can conclude that these generators are different from zero in the corresponding algebra O(S 4n−1 q ).