Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers

Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.

Generating Functionals for Locally Compact Quantum Groups

Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital *-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, symmetric, hermitian conditionally positive functional on a dense *-subalgebra of the unitisation of the universal C$^*$-algebra of a locally compact quantum group, satisfying certain technical conditions, extends in a canonical way to a generating functional. Some consequences of these results are outlined, notably those related to constructing cocycles out of convolution semigroups.

Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

A one-to-one correspondence between shifts of group-like projections on a locally compact quantum group [Formula: see text] which are preserved by the scaling group and contractive idempotent functionals on the dual [Formula: see text] is established. This is a generalization of the Illie–Spronk’s correspondence between contractive idempotents in the Fourier–Stieltjes algebra of a locally compact group [Formula: see text] and cosets of open subgroups of [Formula: see text]. We also establish a one-to-one correspondence between nondegenerate, integrable, [Formula: see text]-invariant ternary rings of operators [Formula: see text], preserved by the scaling group and contractive idempotent functionals on [Formula: see text]. Using our results, we characterize coideals in [Formula: see text] admitting an atom preserved by the scaling group in terms of idempotent states on [Formula: see text]. We also establish a one-to-one correspondence between integrable coideals in [Formula: see text] and group-like projections in [Formula: see text] satisfying an extra mild condition. Exploiting this correspondence, we give examples of group-like projections which are not preserved by the scaling group.

Landstad–Vaes theory for locally compact quantum groups

Landstad–Vaes theory deals with the structure of the crossed product of a [Formula: see text]-algebra by an action of locally compact (quantum) group. In particular, it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of [Formula: see text]-dynamical system introducing the concept of weak action of quantum groups on [Formula: see text]-algebras. It is still possible to define crossed product (by weak action) and characterize the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.

L p {L_{p}} -representations of discrete quantum groups

Given a locally compact quantum group {\mathbb{G}} , we define and study representations and {\mathrm{C}^{\ast}} -completions of the convolution algebra {L_{1}(\mathbb{G})} associated with various linear subspaces of the multiplier algebra {C_{b}(\mathbb{G})} . For discrete quantum groups {\mathbb{G}} , we investigate the left regular representation, amenability and the Haagerup property in this framework. When {\mathbb{G}} is unimodular and discrete, we study in detail the {\mathrm{C}^{\ast}} -completions of {L_{1}(\mathbb{G})} associated with the non-commutative {L_{p}} -spaces {L_{p}(\mathbb{G})} . As an application of this theory, we characterize (for each {p\in[1,\infty)} ) the positive definite functions on unimodular orthogonal and unitary free quantum groups {\mathbb{G}} that extend to states on the {L_{p}} - {\mathrm{C}^{\ast}} -algebra of {\mathbb{G}} . Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups.

Amenability and Covariant Injectivity of Locally Compact Quantum Groups II

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group and 1-injectivity of as an operator -module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra VN(G) is 1-injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded -module maps on , and give a simpliûed proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

Haagerup approximation property via bimodules

The Haagerup approximation property (HAP) is defined for finite von Neumann algebras in such a way that the group von Neumann algebra of a discrete group has the HAP if and only if the group itself has the Haagerup property. The HAP has been studied extensively for finite von Neumann algebras and it was recently generalized to arbitrary von Neumann algebras by Caspers-Skalski and Okayasu-Tomatsu. One of the motivations behind the generalization is the fact that quantum group von Neumann algebras are often infinite even though the Haagerup property has been defined successfully for locally compact quantum groups by Daws-Fima-Skalski-White. In this paper, we fill this gap by proving that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the HAP. This is new even for genuine locally compact groups.

A Homological Property and Arens Regularity of Locally Compact Quantum Groups

We characterize two important notions of amenability and compactness of a locally compact quantum group G in terms of certain homological properties. For this, we show that G is character amenable if and only if it is both amenable and co-amenable. We ûnally apply our results to Arens regularity problems of the quantum group algebra L1(G). In particular, we improve an interesting result by Hu, Neufang, and Ruan.