Compact Group Research Articles

On the growth of Fourier multipliers

We define a sequence of functions, namely, tame cuts, in the Fourier algebra A(G) of a locally compact group G , which satisfies certain convergence and growth conditions. This new consideration allows us to give a group admitting a Fourier multiplier that is not completely bounded. Furthermore, we show that the induction map MA(\Gamma)\rightarrow MA(G) is not always continuous. We also show how Liao’s property (T_{\mathrm{Schur}},G,K) opposes tame cuts. Some examples are provided.

Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows

Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows

Examples of compact quantum groups with [formula omitted] a factor

For each λ∈]0,1] we exhibit an uncountable family of compact quantum groups G such that the von Neumann algebra L∞(G) is the injective factor of type IIIλ with separable predual. We also show that uncountably many injective factors of type III0 arise as L∞(G) for some compact quantum group G. We introduce natural invariants of quantum groups related to the scaling group, modeled on the Connes invariant T for von Neumann algebras. We compute the values of these invariants in our examples, which allows us to distinguish between them. We also investigate their connection to the Connes invariants T(L∞(G)), S(L∞(G)).In the final section we show that for a compact quantum group G the von Neumann algebra L∞(G) cannot have a direct summand of the form B(H) with dim⁡H=∞. In particular, factors of type I (of any dimension) cannot be obtained as L∞(G) for a non-trivial compact quantum group G.

GLOBAL HYPOELLIPTICITY OF SUMS OF SQUARES ON COMPACT MANIFOLDS

Abstract We present necessary and sufficient conditions for an operator of the type sum of squares to be globally hypoelliptic on $T \times G$ , where T is a compact Riemannian manifold and G is a compact Lie group. These conditions involve the global hypoellipticity of a system of vector fields on G and are weaker than Hörmander’s condition, while generalizing the well known Diophantine conditions on the torus. Examples of operators satisfying these conditions in the general setting are provided.

The approximation property for locally compact quantum groups

We study the Haagerup–Kraus approximation property for locally compact quantum groups, generalising and unifying previous work by Kraus–Ruan and Crann. Along the way we discuss how multipliers of quantum groups interact with the C⁎-algebraic theory of locally compact quantum groups. Several inheritance properties of the approximation property are established in this setting, including passage to quantum subgroups, free products of discrete quantum groups, and duals of double crossed products. We also discuss a relation to the weak⁎ operator approximation property. For discrete quantum groups, we introduce a central variant of the approximation property, and relate this to a version of the approximation property for rigid C⁎-tensor categories, building on work of Arano–De Laat–Wahl.

A fixed point theorem for isometries on a metric space

Abstract We show that if X is a complete metric space with uniform relative normal structure and G is a subgroup of the isometry group of X with bounded orbits, then there is a point in X fixed by every isometry in G. As a corollary, we obtain a theorem of U. Lang (2013) concerning injective metric spaces. A few applications of this theorem are given to the problems of inner derivations. In particular, we show that if L 1 ⁢ ( μ ) {L_{1}(\mu)} is an essential Banach L 1 ⁢ ( G ) {L_{1}(G)} -bimodule, then any continuous derivation δ : L 1 ⁢ ( G ) → L ∞ ⁢ ( μ ) {\delta:L_{1}(G)\rightarrow L_{\infty}(\mu)} is inner. This extends a theorem of B. E. Johnson (1991) asserting that the convolution algebra L 1 ⁢ ( G ) {L_{1}(G)} is weakly amenable if G is a locally compact group.

A Simulation of the Dependence of Tidal Interaction on Galaxy Type in Compact Groups

A Simulation of the Dependence of Tidal Interaction on Galaxy Type in Compact Groups

Compact Lie groups and complex reductive groups

Compact Lie groups and complex reductive groups

Density of sums of shifts of a single function in $L_2^0$ space on a compact abelian group

Density of sums of shifts of a single function in $L_2^0$ space on a compact abelian group

Density of the sums of shifts of a single function in the $L_2^0$ space on a compact Abelian group

Density of the sums of shifts of a single function in the $L_2^0$ space on a compact Abelian group

A geometric approach to the Yang-Mills mass gap

I provide a new idea based on geometric analysis to obtain a positive mass gap in pure non-abelian renormalizable Yang-Mills theory. The orbit space, that is the space of connections of Yang-Mills theory modulo gauge transformations, is equipped with a Riemannian metric that naturally arises from the kinetic part of reduced classical action and admits a positive definite sectional curvature. The corresponding regularized Bakry-Émery Ricci curvature (if positive) is shown to produce a mass gap for 2+1 and 3+1 dimensional Yang-Mills theory assuming the existence of a quantized Yang-Mills theory on (ℝ1+2, η) and (ℝ1+3, η), respectively. My result on the gap calculation, described at least as a heuristic one, applies to non-abelian Yang-Mills theory with any compact semi-simple Lie group in the aforementioned dimensions. In 2+1 dimensions, the square of the Yang-Mils coupling constant gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} has the dimension of mass, and therefore the spectral gap of the Hamiltonian is essentially proportional to gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} with proportionality constant being purely numerical as expected. Due to the dimensional restriction on 3+1 dimensional Yang-Mills theory, it seems one ought to introduce a length scale to obtain an energy scale. It turns out that a certain ‘trace’ operation on the infinite-dimensional geometry naturally introduces a length scale that has to be fixed by measuring the energy of the lowest glu-ball state. However, this remains to be understood in a rigorous way.

Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology

Consider a compact group $G$ acting on a real or complex Banach Lie group $U$, by automorphisms in the relevant category, and leaving a central subgroup $K\le U$ invariant. We define the spaces ${}_KZ^n(G,U)$ of $K$-relative continuous cocycles as those maps ${G^n\to U}$ whose coboundary is a $K$-valued $(n+1)$-cocycle; this applies to possibly non-abelian $U$, in which case $n=1$. We show that the ${}_KZ^n(G,U)$ are analytic submanifolds of the spaces $C(G^n,U)$ of continuous maps $G^n\to U$ and that they decompose as disjoint unions of fiber bundles over manifolds of $K$-valued cocycles. Applications include: (a) the fact that ${Z^n(G,U)\subset C(G^n,U)}$ is an analytic submanifold and its orbits under the adjoint of the group of $U$-valued $(n-1)$-cochains are open; (b) hence the cohomology spaces $H^n(G,U)$ are discrete; (c) for unital $C^*$-algebras $A$ and $B$ with $A$ finite-dimensional the space of morphisms $A\to B$ is an analytic manifold and nearby morphisms are conjugate under the unitary group $U(B)$; (d) the same goes for $A$ and $B$ Banach, with $A$ finite-dimensional and semisimple; (e) and for spaces of projective representations of compact groups in arbitrary $C^*$ algebras (the last recovering a result of Martin's).

The Continuous Wavelet Transform in Sobolev Spaces Over Locally Compact Abelian Group and Its Approximation Properties

The Sobolev space over locally compact abelian group Hs(G) is defined and we extend the continuous wavelet transform to Sobolev space Hs(G) for arbitrary real s. This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces. Further, the asymptotic behaviour of the transforms of the L2 function for small scaling parameters is examined. In special cases, the wavelet transform converges to a generalized derivative of its argument. We also discuss the consequences for the discrete wavelet transform arising from this property. Numerical examples illustrate the main result.

Convex-cyclic weighted translations on locally compact groups

A bounded linear operator T on a Banach space X is called a convex-cyclic operator if there exists a vector x ∈ X such that the convex hull of O r b ( T , x ) is dense in X. In this paper, for given an aperiodic element g in a locally compact group G, we give some sufficient conditions for a weighted translation operator T g , w : f ↦ w ⋅ f ∗ δ g on L p ( G ) to be convex-cyclic. A necessary condition is also studied. At the end, to explain the obtained results, some examples are given.

Geometry of Banach algebra [formula omitted] and the bidual of [formula omitted

This article is intended towards the study of the bidual of generalized group algebra L1(G,A) equipped with two Arens products, where G is any locally compact group and A is a Banach algebra. We show that the left topological center of (L1(G)⊗ˆA)⁎⁎ is a Banach L1(G)-module if G is abelian. Further it also holds permanence property with respect to the unitization of A. We then use this fact to extend the remarkable result of A.M. Lau and V. Losert [7], about the topological center of L1(G)⁎⁎ being just L1(G), to the reflexive Banach algebra valued case using the theory of vector measures. We further explore pseudo-center of L1(G,A)⁎⁎ for non-reflexive Banach algebras A and give a partial characterization for elements of pseudo-center using the Cohen's factorization theorem. In the running we also observe few consequences when A holds the Radon-Nikodym property and weak sequential completeness.

Group C*-Algebras of Locally Compact Groups Acting on Trees

Abstract It was proved by Samei and Wiersma that for every non-compact, closed subgroup $G$ of the automorphism group $\textrm {Aut}(T)$ of a (semi-)homogeneous tree $T$ acting transitively on the boundary $\partial T$ and every $2 \leq q < p \leq \infty $, the quotient map $C^{\ast }_{L^{p+}}(G) \twoheadrightarrow C^{\ast }_{L^{q+}}(G)$ is not injective. We prove that whenever $G$, moreover, acts transitively on the vertices of $T$ and has Tits’s independence property, the group $C^{\ast }$-algebras $C^{\ast }_{L^{p+}}(G)$ are the only group $C^{\ast }$-algebras of $G$ coming from ideals of the Fourier–Stieltjes algebra. We also show that given such a group $G$, every group $C^{\ast }$-algebra $C^{\ast }_{\mu }(G)$ that is distinguishable from $C^{\ast }(G)$ and whose dual space $C^{\ast }_{\mu }(G)^{\ast }$ is an ideal in $B(G)$ is abstractly $^{\ast}$-isomorphic to $C^{\ast }_{r}(G)$.

Shading Performance of Public Open Spaces: A Multi-Criteria Evaluation Framework for Housing Projects

This study investigates the impact of building grouping patterns on enhancing shading in public open spaces, considering different solar orientations, housing densities, and the hot climatic conditions of Saudi Arabia. The study considered a set of environmental performance indicators, including the percentage of open space shaded areas, heat gains through the building envelope, open space surface temperature, and the natural ventilation potential of each grouping pattern of the buildings. The results showed that at a building height of five floors, the percentage of open space shaded areas ranged from 14% to 53%. However, using the suggested multi-criteria evaluation framework to consider the above-mentioned performance indicators revealed more in-depth observations. Configurations that included parallel rows of buildings staggered and centralized in the middle of the site offered the best performance in this regard. This shows that compact grouping patterns in which open spaces are decentralized offer more urban shading and protection against undesirable heat gains through the building envelope during summer. The results of this study will enable a wider exploration of different building grouping patterns, which are needed in the harsh hot climate of Saudi Arabia to respond to current rapid housing development plans.

The effect of the stagnant layer on the photon mean-free-path length in concentrated suspensions of nanoparticles

We analyze the possibility of evaluation of the thickness and refractive index of the stagnant layer in concentrated suspensions of nanoparticles through the transport characteristics of scattered light photons. The analysis is based on a physically-transparent generalization of the concept of the single scattering intensity off systems in which the number of particles within regions with linear sizes of order of the wavelength in the medium greatly exceeds unity. This generalization is carried out within the notion of compact groups of particles, makes it possible to go beyond the traditional Born approximation, and take into account many-particle effects contributed from those ranges of integration variables in the terms of the iteration series for the scattered field where the internal propagators have delta-function-type behavior. As a result, the evaluation of the photon transport characteristics becomes possible without a detailed modeling of many-particle scattering and correlation processes in the system. 
 The photon mean-free-path length is investigated theoretically as a function of the stagnant refractive index and that of the layer thickness to show a noticeable effect of both parameters on it. As the layer refractive index is increased at a fixed layer thickness, the mean-free-path length decreases because the suspension optical density increases. As a function of the layer thickness, the photon mean-free-path length reveals different types of behavior, depending on the relation between the refractive indices of the stagnant layer and base liquid. If the former is smaller than the latter, this behavior is increasing; in the opposite case, it is decreasing. An experimentally observed increase of the photon mean-free-path length with the particle concentration is usually explained as a manifestation of higher correlation effects. Our theory reveals that the presence of the stagnant layer make the situation more complicated, for both factors may either enhance or diminish each other. To resolve the issue, new specially-designed experiments are required to be set up.

Non Naturally Reductive Einstein Metrics on the Orthogonal Group via Real Flag Manifolds

We obtain new invariant Einstein metrics on the compact Lie groups SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document} which are not naturally reductive. This is achieved by using the real flag manifolds SO(k1+⋯+kp)/SO(k1)×⋯×SO(kp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(k_1+\\cdots +k_p)/{{\\,\ extrm{SO}\\,}}(k_1)\ imes \\cdots \ imes {{\\,\ extrm{SO}\\,}}(k_p)$$\\end{document} and by imposing certain symmetry assumptions in the set of all left-invariant metrics on SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document}.

Kahane’s Upper Density and Syndetic Sets in LCA Groups

Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.