Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0211002

Haar measure on a locally compact quantum group

Byung-Jay Kahng
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the general theory of locally compact quantum groups, the notion of Haar measure (Haar weight) plays the most significant role. The aim of this paper is to carry out a careful analysis regarding Haar weight, in relation to general theory, for the specific non-compact quantum group (A,Δ) constructed earlier by the author. In this way, one can show that (A,Δ) is indeed a ``(C*-algebraic) locally compact quantum group'' in the sense of the recently developed definition given by Kustermans and Vaes. Attention will be given to pointing out the relationship between the original construction (obtained by deformation quantization) and the structure maps suggested by general theory.

Explore related subjects

math.OAmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Byung-Jay Kahng. 2004-05-05. Haar measure on a locally compact quantum group. https://arxiv.org/abs/math/0211002

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rings of non-commutative functions and their fields of fractions

Semi-free ideal rings, or semifirs, were introduced by Paul M. Cohn to study universal localizations in the non-commutative setting. We provide new examples of semifirs consisting of analytic functions in several non-commuting variables. These examples arise canonically in free analysis by completing the free algebra in the topology of ``uniform convergence on operator-space balls'' in the non-commutative universe of tuples of square matrices of any finite size. We show, in particular, that the ring of (uniformly) entire non-commutative (NC) functions in $d \in \mathbb{N}$ non-commuting variables, $\scr{O}_d$, is a semifir. Every finitely--generated right (or left) ideal in $\scr{O}_d$ is closed, which yields an analytic extension of G. Bergman's nullstellensatz for the free algebra. Any semifir admits a universal skew field of fractions; applying this to $\scr{O}_d$ yields the universal skew field of ``NC meromorphic expressions", $\scr{M} _d$. We show that any $f \in \scr{M} _d$ has a well-defined domain and evaluations in a large class of stably-finite topological algebras, including finite $C^*$-algebras, extending a result of Cohn for NC rational functions. As an application, we extend the almost sure convergence result of Haagerup and Thorbjörnsen for free polynomials evaluated on tuples of random matrices to the setting of NC meromorphic expressions.

math.OA↗
arXiv2026

Quantum channels on duals of von Neumann algebras in the Schrödinger picture

The theory of quantum channels is traditionally studied either on finite-dimensional state spaces or within the Heisenberg picture as completely positive maps on C^*-algebras. In this paper, we consider quantum channels as completely positive maps on the duals of general von Neumann algebras in the Schrodinger picture. We investigate the construction of such channels through Pettis integrals using representations of topological groups.

math.OA↗
arXiv2026

Infinitesimal Freeness of Wigner Matrices

In this paper, within the framework of real infinitesimal free probability introduced by Cébron and the second author, we compute the real infinitesimal free cumulants of independent complex Wigner matrices. Our approach relies on establishing a combinatorial relation between annular non-crossing partitions and families of directed graphs. As a consequence, we demonstrate that independent complex Wigner matrices are asymptotically real infinitesimally free. In particular, we show (under mild conditions) that a complex Wigner matrix is asymptotically infinitesimally free from its transpose.

math.OA↗