S. L. Woronowicz's theory of C*-algebras generated by unbounded elements is applied to q-normal operators satisfying the defining relation of the quantum complex plane. The unique non-degenerate C*-algebra of bounded operators generated by a q-normal operator is computed and an abstract description is given by using crossed product algebras. If the spectrum of the modulus of the q-normal operator is the positive half line, this C*-algebra will be considered as the algebra of continuous functions...

We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of forming operators from non-commuting symbols can be considered as a second quantization. To do this we construct a reproducing kernel associated with the quantum plane. We also discuss the commutation relations of creation and annihilation operators which are defined...

Let G be a locally compact group, and consider the weakly almost periodic functionals on M(G), the measure algebra of G, denoted by WAP(M(G)). This is a C*-subalgebra of the commutative C*-algebra M(G)*, and so has character space, say $K_{WAP}$. In this paper, we investigate properties of $K_{WAP}$. We present a short proof that $K_{WAP}$ can naturally be turned into a semigroup whose product is separately continuous; at the Banach algebra level, this product is simply the natural one induced by the Arens products. This...

The non-commutative torus $C^{*}({\mathbb{Z}}^n,\omega )$ is realized as the $C^{*}$-algebra of sections of a locally trivial $C^{*}$-algebra bundle over $\widehat{S_{\omega }}$ with fibres isomorphic to $C^{*}({\mathbb{Z}}^n/S_{\omega },{\omega }_1)$ for a totally skew multiplier ${\omega }_1$ on ${\mathbb{Z}}^n/S_{\omega }$. D. Poguntke [9] proved that $A_{\omega }$ is stably isomorphic to $C\left(\widehat{S_{\omega }}\right)\otimes C^{*}({\mathbb{Z}}^n/S_{\omega },{\omega }_1)\cong C\left(\widehat{S_{\omega }}\right)\otimes A_{\varphi }\otimes M_{kl}\left(ℂ\right)$ for a simple non-commutative torus $A_{\varphi }$ and an integer $kl$. It is well-known that a stable isomorphism of two separable $C^{*}$-algebras is equivalent to the existence of equivalence bimodule between them. We construct an $A_{\omega }$-$C\left(\widehat{S_{\omega }}\right)\otimes A_{\varphi }$-equivalence bimodule.

The half-liberated orthogonal group $O_n^{*}$ appears as intermediate quantum group between the orthogonal group $O_n$, and its free version $O_n^{+}$. We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between $O_n^{*}$ and $U_n$, a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that...

We show that the stable C*-algebra and the related Ruelle algebra defined by I. Putnam from the irreducible Smale space associated with a topologically mixing expanding map of a compact metric space are strongly Morita equivalent to the groupoid C*-algebras defined directly from the expanding map by C. Anantharaman-Delaroche and V. Deaconu. As an application, we calculate the K⁎-group of the Ruelle algebra for a solenoid.

In a recent article, Kenny De Commer investigated Morita equivalence between locally compact quantum groups, in which a measured quantum groupoid, of basis ℂ², was constructed as a linking object. Here, we generalize all these constructions and concepts to the level of measured quantum groupoids. As for locally compact quantum groups, we apply this construction to the deformation of a measured quantum groupoid by a 2-cocycle.

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in...