This dissertation is based on several projects in the area of operator algebras, inparticular graph C∗-algebras, and noncommutative geometry.The first project we present lies completely in the area of graph C∗-algebras inwhich we study inner versus outer conjugacy of subalgebras of graph C∗-algebras.Let E be a finite graph, for the canonical diagonal maximal abelian subalgebra DE ofC∗(E) we show that if α is a vertex-fixing quasi-free automorphism of C∗(E) for whichα(DE) 6= DE then α(DE) and DE are not inner conjugate. Furthermore, we present acriterion which guarantees that a polynomial automorphism moves the canonical UHFsubalgebra Fn of the Cuntz algebra to a non-inner conjugate UHF subalgebra. Thisbased on joint work with Tomohiro Hayashi, Jeong Hee Hong and Wojciech Szymanski.
In the other projects, we study quantum analogues of classical spaces and noncommutative bundles. Motivated by the equivalence between the categories of commutative C∗-algebras and continuous functions on locally compact spaces, due to Gelfand,we often think about noncommutative C∗-algebras as continuous functions on a nonexisting virtual quantum space. For many classical spaces a quantum analogue is thengiven as a noncommutative C∗-algebra.In joint work with Thomas Gotfredsen, we investigate classification of quantumanalogues of the classical lens spaces for dimension at most 7, which can be viewedas graph C∗-algebras. Applying the classification result of finite graphs by Eilers,Restorff, Ruiz and Sørensen we find a number-theoretic invariant of the associatedC∗-algebra.Another noncommutative space of interest is the quantum complex projective spaceC(CPnq) by Vaksman and Soibelman. In joint work with Francesca Arici we present anexplicit KK-equivalence between C(CPnq) and Cn+1. In the construction it is crucialthat C(CPnq) can be viewed as a graph C∗-algebra, as shown by Hong and Szymanski.
The theory of noncommutative principal bundles is well-established by now, butnot much is known about general fibre bundles. Recently Brzezi´nski and Szymanskipresented a description of noncommutative bundles with homogeneous fibres at thepurely algebraic level. As an example of this, a quantum twistor bundle is constructed.It is defined by passing to the fixed-point algebras of a U(1) action on the quantuminstanton bundle by Bonechi, Ciccoli, Dabrowski and Tarlini. In a joint project withWojciech Szyma´nski, we investigate the enveloping C∗-algebras of the algebras in thebundle. It is shown that the enveloping C∗-algebra of the total space of the quantumtwistor bundle is isomorphic to the well known C(CP3q) by Vaksman and Soibelman.In the light of this, we moreover consider another quantum instanton bundle definedby Landi, Pagani and Reina. We investigate the enveloping C∗-algebra of the totalspace of the bundle, namely the symplectic 7-sphere. It is shown that one of thegenerators is forced to be zero in the C∗-algebra. This result was later generalisedby Landi and D’Andrea to general quantum symplectic spheres C(S4n−1q). Moreover,we prove that C(S4n−1q) is isomorphic to the well-studied quantum (2n + 1)-sphere byVaksman and Soibelman and a vector space basis of its polynomial dense ∗-subalgebrais constructed.