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1. CMB Online first

Pachl, Jan; Steprāns, Juris
 Continuity of convolution and SIN groups Let the measure algebra of a topological group $G$ be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly continuous if and only if $G$ has the SIN property. On the larger space $\mathsf{LUC}(G)^\ast$ which includes the measure algebra, convolution is also jointly continuous if and only if the group has the SIN property, but not separately continuous for many non-SIN groups. Keywords:topological group, SIN property, measure algebra, convolutionCategories:43A10, 22A10

2. CMB Online first

Alaghmandan, Mahmood; Crann, Jason
 Character density in central subalgebras of compact quantum groups We investigate quantum group generalizations of various density results from Fourier analysis on compact groups. In particular, we establish the density of characters in the space of fixed points of the conjugation action on $L^2(\mathbb{G})$, and use this result to show the weak* density and norm density of characters in $ZL^\infty(\mathbb{G})$ and $ZC(\mathbb{G})$, respectively. As a corollary, we partially answer an open question of Woronowicz. At the level of $L^1(\mathbb{G})$, we show that the center $\mathcal{Z}(L^1(\mathbb{G}))$ is precisely the closed linear span of the quantum characters for a large class of compact quantum groups, including arbitrary compact Kac algebras. In the latter setting, we show, in addition, that $\mathcal{Z}(L^1(\mathbb{G}))$ is a completely complemented $\mathcal{Z}(L^1(\mathbb{G}))$-submodule of $L^1(\mathbb{G})$. Keywords:compact quantum group, irreducible characterCategories:43A20, 43A40, 46J40

3. CMB Online first

Chen, Chung-Chuan
 Disjoint hypercyclicity and weighted translations on discrete groups Let $1\leq p\lt \infty$, and let $G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space $\ell^p(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained. Keywords:disjoint hypercyclicity, topological transitivity, weighted translation, $\ell^p$-spaceCategories:47A16, 47B38, 43A15

4. CMB Online first

Shravan Kumar, N.
 Invariant means on a class of von Neumann Algebras related to Ultraspherical Hypergroups II Let $K$ be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi$ and let $VN(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K.$ In this note, we show that the set of invariant means on $VN(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$ in the $cb$-multiplier norm. Finally, we consider generalized translations and generalized invariant means. Keywords:ultraspherical hypergroup, Fourier algebra, Fourier-Stieltjes algebra, invariant mean, generalized translation, generalized invariant meanCategories:43A62, 46J10, 43A30, 20N20

5. CMB 2016 (vol 60 pp. 122)

Ghanei, Mohammad Reza; Nasr-Isfahani, Rasoul; Nemati, Mehdi
 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 ${\mathbb G}$ in terms of certain homological properties. For this, we show that ${\mathbb G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra $L^1({\mathbb G})$; in particular, we improve an interesting result by Hu, Neufang and Ruan. Keywords:amenability, Arens regularity, co-amenability, locally compact quantum group, homological propertyCategories:46L89, 43A07, 46H20, 46M10, 58B32

6. CMB 2016 (vol 60 pp. 111)

Ghaani Farashahi, Arash
 Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the normalized $G$-invariant measure on $G/H$ associated to the Weil's formula. Then, we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space $L^2(G/H,\mu)$. Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formulaCategories:20G05, 43A85, 43A32, 43A40

7. CMB 2016 (vol 59 pp. 528)

Jahan, Qaiser
 Characterization of Low-pass Filters on Local Fields of Positive Characteristic In this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field $K$ of positive characteristic associated to the scaling function for multiresolution analysis of $L^2(K)$. We use probability and martingale methods to provide such a characterization. Keywords:multiresolution analysis, local field, low-pass filter, scaling function, probability, conditional probability and martingalesCategories:42C40, 42C15, 43A70, 11S85

8. CMB 2016 (vol 59 pp. 693)

Chen, Chung-Chuan
 Recurrence of Cosine Operator Functions on Groups In this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent. Keywords:topologically multiple recurrence, recurrence, topological transitivity, hypercyclicity, cosine operator functionCategories:47A16, 54B20, 43A15

9. CMB 2016 (vol 59 pp. 521)

Hare, Kathryn; Ramsey, L. Thomas
 The Relationship Between $\epsilon$-Kronecker Sets and Sidon Sets A subset $E$ of a discrete abelian group is called $\epsilon$-Kronecker if all $E$-functions of modulus one can be approximated to within $\epsilon$ by characters. $E$ is called a Sidon set if all bounded $E$-functions can be interpolated by the Fourier transform of measures on the dual group. As $% \epsilon$-Kronecker sets with $\epsilon \lt 2$ possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true. Keywords:Kronecker set, Sidon setCategories:43A46, 42A15, 42A55

10. CMB 2015 (vol 58 pp. 632)

Silberman, Lior
 Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift Given a measure $\bar\mu_\infty$ on a locally symmetric space $Y=\Gamma\backslash G/K$, obtained as a weak-{*} limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure $\bar\mu_\infty$ on the homogeneous space $X=\Gamma\backslash G$ which lifts $\bar\mu_\infty$ and which is invariant by a connected subgroup $A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then $\bar\mu_\infty$ is also the limit of measures associated to Hecke eigenfunctions on $X$. This generalizes results of the author with A. Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber. Keywords:quantum unique ergodicity, microlocal lift, spherical dualCategories:22E50, 43A85

11. CMB 2015 (vol 58 pp. 415)

Willson, Benjamin
 A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability In this paper we present a fixed point property for amenable hypergroups which is analogous to Rickert's fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, a certain class of hypergroups are shown to have a left Haar measure. Keywords:invariant measure, Haar measure, hypergroup, amenability, function translationsCategories:43A62, 43A05, 43A07

12. CMB 2014 (vol 58 pp. 3)

Alaghmandan, Mahmood
 Approximate Amenability of Segal Algebras II We prove that every proper Segal algebra of a SIN group is not approximately amenable. Keywords:Segal algebras, approximate amenability, SIN groups, commutative Banach algebrasCategories:46H20, 43A20

13. CMB 2014 (vol 57 pp. 834)

Koh, Doowon
 Restriction Operators Acting on Radial Functions on Vector Spaces Over Finite Fields We study $L^p-L^r$ restriction estimates for algebraic varieties $V$ in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties $V$ lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties $V$ are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions. Keywords:finite fields, radial functions, restriction operatorsCategories:42B05, 43A32, 43A15

14. CMB 2013 (vol 57 pp. 449)

Alaghmandan, Mahmood; Choi, Yemon; Samei, Ebrahim
 ZL-amenability Constants of Finite Groups with Two Character Degrees We calculate the exact amenability constant of the centre of $\ell^1(G)$ when $G$ is one of the following classes of finite group: dihedral; extraspecial; or Frobenius with abelian complement and kernel. This is done using a formula which applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk (J. Funct. Anal. 2009). Keywords:center of group algebras, characters, character degrees, amenability constant, Frobenius group, extraspecial groupsCategories:43A20, 20C15

15. CMB 2013 (vol 56 pp. 729)

Currey, B.; Mayeli, A.
 The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is a unitary representation of a wavelet group and $\Gamma$ is the abstract pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the form $\tau(\Gamma)\Psi$ where $\tau$ is a super-representation of $\pi$. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications. Keywords:frame, dilation, wavelet, Baumslag-Solitar group, shearletCategories:43A65, 42C40, 42C15

16. CMB 2012 (vol 57 pp. 289)

Ghasemi, Mehdi; Marshall, Murray; Wagner, Sven
 Closure of the Cone of Sums of $2d$-powers in Certain Weighted $\ell_1$-seminorm Topologies In a paper from 1976, Berg, Christensen and Ressel prove that the closure of the cone of sums of squares $\sum \mathbb{R}[\underline{X}]^2$ in the polynomial ring $\mathbb{R}[\underline{X}] := \mathbb{R}[X_1,\dots,X_n]$ in the topology induced by the $\ell_1$-norm is equal to $\operatorname{Pos}([-1,1]^n)$, the cone consisting of all polynomials which are non-negative on the hypercube $[-1,1]^n$. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $\ell_1$-seminorm topology associated to an absolute value. In the present paper we give a new proof of these results which is based on Jacobi's representation theorem from 2001. At the same time, we use Jacobi's representation theorem to extend these results from sums of squares to sums of $2d$-powers, proving, in particular, that for any integer $d\ge 1$, the closure of the cone of sums of $2d$-powers $\sum \mathbb{R}[\underline{X}]^{2d}$ in $\mathbb{R}[\underline{X}]$ in the topology induced by the $\ell_1$-norm is equal to $\operatorname{Pos}([-1,1]^n)$. Keywords:positive definite, moments, sums of squares, involutive semigroupsCategories:43A35, 44A60, 13J25

17. CMB 2012 (vol 57 pp. 37)

Dashti, Mahshid; Nasr-Isfahani, Rasoul; Renani, Sima Soltani
 Character Amenability of Lipschitz Algebras Let ${\mathcal X}$ be a locally compact metric space and let ${\mathcal A}$ be any of the Lipschitz algebras ${\operatorname{Lip}_{\alpha}{\mathcal X}}$, ${\operatorname{lip}_{\alpha}{\mathcal X}}$ or ${\operatorname{lip}_{\alpha}^0{\mathcal X}}$. In this paper, we show, as a consequence of rather more general results on Banach algebras, that ${\mathcal A}$ is $C$-character amenable if and only if ${\mathcal X}$ is uniformly discrete. Keywords:character amenable, character contractible, Lipschitz algebras, spectrumCategories:43A07, 46H05, 46J10

18. CMB 2011 (vol 56 pp. 13)

 Ordering the Representations of $S_n$ Using the Interchange Process Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett, and Richthammer, we define an associated order $\prec$ on the irreducible representations of $S_n$. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the Aldous order'' completely is a generalized question. We show a few additional entries for this order. Keywords:Aldous' conjecture, interchange process, symmetric group, representationsCategories:82C22, 60B15, 43A65, 20B30, 60J27, 60K35

19. CMB 2011 (vol 56 pp. 218)

Yang, Dilian
 Functional Equations and Fourier Analysis By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations - the d'Alembert equation, the Wilson equation, and the d'Alembert long equation - on compact groups. Keywords:functional equations, Fourier analysis, representation of compact groupsCategories:39B52, 22C05, 43A30

20. CMB 2011 (vol 54 pp. 654)

Forrest, Brian E.; Runde, Volker
 Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.) Keywords:amenability, bounded approximate identity, $cb$-multiplier norm, Fourier algebra, norm one idempotentCategories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25

21. CMB 2011 (vol 54 pp. 663)

Haas, Ruth; G. Helminck, Aloysius
 Admissible Sequences for Twisted Involutions in Weyl Groups Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$ related to a basis $\Delta$ for the root system $\Phi$ associated with $W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We show that the set of $\theta$-twisted involutions in $W$, $\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{-1}\}$ is in one to one correspondence with the set of regular involutions $\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are characterized by sequences in $\Sigma$ which induce an ordering called the Richardson-Springer Poset. In particular, for $\Phi$ irreducible, the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$, for nontrivial $\theta$ is identical to the descending Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$. Categories:20G15, 20G20, 22E15, 22E46, 43A85

22. CMB 2011 (vol 54 pp. 544)

Strungaru, Nicolae
 Positive Definite Measures with Discrete Fourier Transform and Pure Point Diffraction In this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups. Keywords:pure point diffraction, positive definite measure, Fourier transform of measuresCategory:43A25

23. CMB 2010 (vol 54 pp. 207)

Chen, Jiecheng; Fan, Dashan
 A Bilinear Fractional Integral on Compact Lie Groups As an analog of a well-known theorem on the bilinear fractional integral on $\mathbb{R}^{n}$ by Kenig and Stein, we establish the similar boundedness property for a bilinear fractional integral on a compact Lie group. Our result is also a generalization of our recent theorem about the bilinear fractional integral on torus. Keywords:bilinear fractional integral, $L^p$ spaces, Heat kernelCategories:43A22, 43A32, 43B25

24. CMB 2010 (vol 54 pp. 126)

Jin, Yongyang; Zhang, Genkai
 Fundamental Solutions of Kohn Sub-Laplacians on Anisotropic Heisenberg Groups and H-type Groups We prove that the fundamental solutions of Kohn sub-Laplacians $\Delta + i\alpha \partial_t$ on the anisotropic Heisenberg groups are tempered distributions and have meromorphic continuation in $\alpha$ with simple poles. We compute the residues and find the partial fundamental solutions at the poles. We also find formulas for the fundamental solutions for some matrix-valued Kohn type sub-Laplacians on H-type groups. Categories:22E30, 35R03, 43A80

25. CMB 2010 (vol 54 pp. 3)

Bakonyi, M.; Timotin, D.
 Extensions of Positive Definite Functions on Amenable Groups Let $S$ be a subset of an amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of this paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as corollaries. New applications are also presented. Categories:43A35, 47A57, 20E05
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