Covariant representations of subproduct systems

A celebrated theorem of Pimsner states that a covariant representation $T$ of a $C^*$-correspondence $E$ extends to a $C^*$-representation of the Toeplitz algebra of $E$ if and only if $T$ is isometric. This paper is mainly concerned with finding conditions for a covariant representation of a \emph{subproduct system} to extend to a $C^*$-representation of the Toeplitz algebra. This framework is much more general than the former. We are able to find sufficient conditions, and show that in important special cases, they are also necessary. Further results include the universality of the tensor algebra, dilations of completely contractive covariant representations, Wold decompositions and von Neumann inequalities.Comment: 43 pages. Incorporates a few minor revisions, suggested by the referee and other

Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms

The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a L\'{e}vy process on a locally compact group. In particular a definitive one-to-one correspondence between symmetric convolution semigroups of states and noncommutative Dirichlet forms satisfying the natural translation invariance property is established, extending earlier partial results and providing a powerful tool to analyse such semigroups. This is then applied to provide new characterisations of the Haagerup Property and Property (T) for locally compact quantum groups, and some examples are presented. The proofs of the main theorems require developing certain general results concerning Haagerup's $L^{p}$-spaces.Comment: 52 pages. v2: minor changes. To appear in Journal de Math\'ematiques Pures et Appliqu\'ee

Ergodic theory for quantum semigroups

Recent results of L. Zsido, based on his previous work with C. P. Niculescu and A. Stroh, on actions of topological semigroups on von Neumann algebras, give a Jacobs-de Leeuw-Glicksberg splitting theorem at the von Neumann algebra (rather than Hilbert space) level. We generalize this to the framework of actions of quantum semigroups, namely Hopf-von Neumann algebras. To this end, we introduce and study a notion of almost periodic vectors and operators that is suitable for our setting.Comment: 21 pages. v2: minor changes. To appear in the Journal of the London Mathematical Societ

On positive definiteness over locally compact quantum groups

The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.Comment: 28 pages; v3: incorporated several changes, most at the referee's suggestion; to appear in the Canadian Journal of Mathematic

Around Property (T) for Quantum Groups

We study Property (T) for locally compact quantum groups, providing several new characterisations, especially related to operator algebraic ergodic theory. Quantum Property (T) is described in terms of the existence of various Kazhdan type pairs, and some earlier structural results of Kyed, Chen and Ng are strengthened and generalised. For second countable discrete unimodular quantum groups with low duals, Property (T) is shown to be equivalent to Property (T)1,1 of Bekka and Valette. This is used to extend to this class of quantum groups classical theorems on ‘typical’ representations (due to Kerr and Pichot), and on connections of Property (T) with spectral gaps (due to Li and Ng) and with strong ergodicity of weakly mixing actions on a particular von Neumann algebra (due to Connes and Weiss). Finally, we discuss in the Appendix equivalent characterisations of the notion of a quantum group morphism with dense image

Convolution semigroups on Rieffel deformations of locally compact quantum groups

Consider a locally compact quantum group $\mathbb{G}$ with a closed classical abelian subgroup $\Gamma$ equipped with a $2$-cocycle $\Psi:\hat{\Gamma}\times\hat{\Gamma}\to\mathbb{C}$. We study in detail the associated Rieffel deformation $\mathbb{G}^{\Psi}$ and establish a canonical correspondence between $\Gamma$-invariant convolution semigroups of states on $\mathbb{G}$ and on $\mathbb{G}^{\Psi}$.Comment: 36 pages. v2: minor corrections. Comments are welcome

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by radical relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C*-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the weak-operator closures of these algebras as well.Comment: 46 pages. Final version, to appear in Advances in Mathematic