Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0005225

On the Quantum Quarter Plane and the Real Quantum Plane

Konrad Schmuedgen
View original source ↗Read source document ↗Export metadata ↓

Abstract

Suppose q is a complex number of modulus one and different from 1,-1. Let O(R^2_q) be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using operator representations of the *-algebra O(R^2_q) on Hilbert space and the Weyl calculus of pseudodifferential operators we construct *-algebras of ``functions'' on the quantum quarter plane R^{++}_q and on the real quantum plane R^2_q which are left module *-algebras for the Hopf *-algebra U_q(gl_2(R)). We define a family h_k, k\in Z^2, of covariant positive linear functionals on these *-algebras and study the actions of the *-algebras O(R^2_q) and U_q(gl_2(R)) on the associated Hilbert spaces. Quantum analogs of the partial Fourier transforms and the Fourier transform are found. A differential calculus on the ``function'' *-algebras is also developed and investigated. A shorter version appeared in International Journal of Mathematics 13 (2002), 279-321.

Explore related subjects

math.OAmath.QA

Keep this discovery

BibTeXRIS

Konrad Schmuedgen. 2000-05-23. On the Quantum Quarter Plane and the Real Quantum Plane. https://arxiv.org/abs/math/0005225

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

KEEP EXPLORING

Related papers

arXiv2026

On the II$_{1}$ Factors of Fuchsian Groups

We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group factor: $L(\mathbb{F}_{2})=W^{*}(w)*W^{*}(v)$ for some Haar unitary $v\in L(\mathbb{F}_{2})$ that is freely independent from $w$. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$, $L(\Gamma)$ is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.

math.OA↗
arXiv2026

On AF- and type I-ideals in certain crossed product C$^\ast$-algebras

We study locally finite-dimensional ideals in crossed products of totally disconnected spaces by free actions of the integers and in uniform Roe algebras of exact discrete groups. In the first case, we present a dynamical description of the largest locally finite-dimensional ideal, which turns out to coincide with the intersection of all maximal ideals. In the latter case, we provide a coarse geometric characterization of the locally finite-dimensional compact ideals. Moreover, we show that for crossed products of totally disconnected spaces by free actions of exact groups, the largest type I-ideal is locally finite-dimensional. In the case of uniform Roe algebras, we provide coarse geometric conditions for compact ideals guaranteeing that the ideal is type I and admits an embedding of a UHF-algebra, respectively.

math.OA↗
arXiv2026

Continuous family of compact quantum metric space structures from cocycle twisted crossed product $\textrm{C}^{\ast}$-algebras

We establish the existence of a three-parameter family of compact quantum metric space structures on cocycle twisted crossed products by discrete groups. We are mainly interested in the case where the acting group has exponential/subexponential growth. We prove that the family is jointly continuous with respect to the parameters when the acting group is exact. We obtain quantitative upper and lower bounds for the associated metric dimensions. In particular, the bounds are helpful to prove the failure of lower semicontinuity of the metric dimension with respect to the quantum Gromov-Hausdorff distance. We also prove invariance of metric dimension under zero quantum Gromov-Hausdorff distance.

math.OA↗