A. van Daele

activity

19962024

most citedLocally Compact Quantum Groups. A von Neumann Algebra Approach

72 citations · 139 across the 19 of their papers we have counts for

collaborators

23 papers

math.RA2024

Single sided multiplier Hopf algebras

Alfons Van Daele

Let $A$ be a non-degenerate algebra over the complex numbers and $Δ$ a homomorphism from $A$ to the multiplier algebra $M (A \otimes A)$ . Consider the linear maps $T_{1}$ and $T_{2}$ fro…

math.RA2024

Reflections on coproducts for non-unital algebras

Alfons Van Daele

A coproduct on a vector space $A$ is defined as a linear map $Δ : A \to A \otimes A$ satisfying coassociativity $(Δ \otimes ι) Δ = (ι \otimes Δ) Δ$ . We use $ι$ for the identity map. If $G$ is…

math.FA2023

Polynomial functions for locally compact group actions

Magnus B. Landstad, Alfons Van Daele

Consider a locally compact group $G$ and a locally compact space $X$ . A local right action of $G$ on $X$ is a continuous map $(x, p) \mapsto x \cdot p$ from an open subset $Γ$ of the…

math.QA2023

Algebraic quantum groups III. The modular and the analytic structure

Alfons Van Daele

This is the last part of a series of three papers on the subject. In the first part we have considered the duality of algebraic quantum groups. In that paper, we use the term algeb…

math.QA2023

Algebraic quantum groups and duality II. Multiplier Hopf *-algebras with positive integrals

Alfons Van Daele

In the paper Algebraic quantum groups and duality I, we consider a pairing $(a, b) \mapsto ⟨ a, b ⟩$ of regular multiplier Hopf algebras $A$ and $B$ . When $A$ has integrals…

math.QA2023

Algebraic quantum groups and duality I

Alfons Van Daele

Let $(A, Δ)$ be a finite-dimensional Hopf algebra. The linear dual $B$ of $A$ is again a finite-dimensional Hopf algebra. The duality is given by an element $V \in B \otimes A$ , defin…

math.RA2019

Pairing and duality of algebraic quantum groupoids

T. Timmermann, A. Van Daele, S. H. Wang

Algebraic quantum groupoids have been developed by two of the authors (AVD and SHW) of this note in a series of papers. Regular multiplier Hopf algebroids are obtained also by two…

math.OA2017★ 4 cited

A class of C*-algebraic locally compact quantum groupoids Part II: Main theory

Byung-Jay Kahng, Alfons Van Daele

This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids ("quantum groupoids of separable type"), based on the pu…

math.RA2017★ 6 cited

Weak multiplier Hopf algebras III. Integrals and duality

Alfons Van Daele, Shuanhong Wang

Let $(A, Δ)$ be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra $A$ , with or without identity, and a coproduct $Δ$ on $A$ , satisfying certain properties. Th…

math.QA2016

Separability idempotents in C*-algebras

Byung-Jay Kahng, Alfons Van Daele

In this paper, we study the notion of a separability idempotent in the C*-algebra framework. This is analogous to the notion in the purely algebraic setting, typically considered i…

math.RA2014★ 3 cited

Modified weak multiplier Hopf algebras

Alfons Van Daele

Let $(A, Δ)$ be a regular weak multiplier Hopf algebra. Denote by $E$ the canonical idempotent of $(A, Δ)$ and by $B$ the image of the source map. Recall that $B$ is a non-degenerate…

math.RA2014

Multiplier Hopf algebroids arising from weak multiplier Hopf algebras

Thomas Timmermann, Alfons Van Daele

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the noti…

math.RA2014

The Larson-Sweedler theorem for weak multiplier Hopf algebras

Byung-Jay Kahng, Alfons Van Daele

The Larson-Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra. The result has been generalized to finite-dimensional weak Hopf alg…

math.RA2014★ 16 cited

Weak Multiplier Hopf Algebras II. The source and target algebras

Alfons Van Daele, Shuanhong Wang

In this paper, we continue the study of weak multiplier Hopf algebras. We recall the notions of the source and target maps $ε_{s}$ and $ε_{t}$ , as well as of the so…

math.QA2013★ 1 cited

Multiplier Hopf algebroids. Basic theory and examples

Thomas Timmermann, Alfons Van Daele

Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable b…

math.RA2013★ 13 cited

Separability Idempotents and Multiplier Algebras

Alfons Van Daele

Consider two non-degenerate algebras B and C over the complex numbers. We study a certain class of idempotent elements E in the multiplier algebra of the tensor product of B with C…

math.RA2012★ 1 cited

Bicrossproducts of algebraic quantum groups

Lydia Delvaux, Alfons Van Daele, Shuanhong Wang

Let $A$ and $B$ be two algebraic quantum groups (i.e. multiplier Hopf algebras with integrals). Assume that $B$ is a right $A$ -module algebra and that $A$ is a left $B$ -comodule co…

math.RA2010

Algebraic quantum groups II. Constructions and examples

L. Delvaux, A. Van Daele

Let G be a group and let A be the algebra of complex functions on G with finite support. The product in G gives rise to a coproduct on A making it a multiplier Hopf algebra. In fac…

math.RA2006★ 2 cited

A note on Radford's $S^{4}$ formula

L. Delvaux, A. Van Daele, Shuanhong Wang

In this note, we show that Radford's formula for the fourth power of the antipode can be proven for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). T…

math.OA2006★ 72 cited

Locally Compact Quantum Groups. A von Neumann Algebra Approach

Alfons Van Daele

In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We develop the theory completely within the von…

funct-an1997★ 14 cited

Universal C*-algebraic quantum groups arising from algebraic quantum groups

Johan Kustermans

In this paper, we construct a universal C*-algebraic quantum group out of an algebraic one. We show that this universal C*-algebraic quantum group has the same rich structure as it…

q-alg1996

C^*-algebraic quantum groups arising from algebraic quantum groups

Johan Kustermans, Alfons Van Daele

We associate to an algebraic quantum group a C^*-algebraic quantum group and prove that this C^*-algebraic quantum group satisfies an upcoming definition of Masuda, Nakagami & Woro…

q-alg1996★ 7 cited

Pairing and Quantum Double of Multiplier Hopf Algebras

Bernhard Drabant, Alfons Van Daele

We define and investigate pairings of multiplier Hopf algebras. It is shown that two dually paired regular multiplier Hopf ( $*$ -)algebras $A$ and $B$ yield a quantum double multipl…

Single sided multiplier Hopf algebras

Reflections on coproducts for non-unital algebras

Polynomial functions for locally compact group actions

Algebraic quantum groups III. The modular and the analytic structure

Algebraic quantum groups and duality II. Multiplier Hopf *-algebras with positive integrals

Algebraic quantum groups and duality I

Pairing and duality of algebraic quantum groupoids

A class of C*-algebraic locally compact quantum groupoids Part II: Main theory

Weak multiplier Hopf algebras III. Integrals and duality

Separability idempotents in C*-algebras

Modified weak multiplier Hopf algebras

Multiplier Hopf algebroids arising from weak multiplier Hopf algebras

The Larson-Sweedler theorem for weak multiplier Hopf algebras

Weak Multiplier Hopf Algebras II. The source and target algebras

Multiplier Hopf algebroids. Basic theory and examples

Separability Idempotents and Multiplier Algebras

Bicrossproducts of algebraic quantum groups

Algebraic quantum groups II. Constructions and examples

A note on Radford's S4 formula

Locally Compact Quantum Groups. A von Neumann Algebra Approach

Universal C*-algebraic quantum groups arising from algebraic quantum groups

C^*-algebraic quantum groups arising from algebraic quantum groups

Pairing and Quantum Double of Multiplier Hopf Algebras

A note on Radford's $S^{4}$ formula