Recent zbMATH articles in MSC 43Ahttps://www.zbmath.org/atom/cc/43A2021-04-16T16:22:00+00:00Werkzeug\(L^1\)-determined primitive ideals in the \(C^\ast\)-algebra of an exponential Lie group with closed non-\(\ast\)-regular orbits.https://www.zbmath.org/1456.430012021-04-16T16:22:00+00:00"Inoue, Junko"https://www.zbmath.org/authors/?q=ai:inoue.junko"Ludwig, Jean"https://www.zbmath.org/authors/?q=ai:ludwig.jeanSummary: Let \(G = \exp (\mathfrak{g})\) be an exponential solvable Lie group and Ad\((G) \subset \mathbb{D}\) an exponential solvable Lie group of automorphisms of \(G\). Assume that for every non-\(\ast\)-regular orbit \(\mathbb{D} \cdot q$, $q \in \mathfrak{g}^\ast\), of \(\mathbb{D} = \exp(\mathfrak{d})\) in \(g^\ast\), there exists a nilpotent ideal \(\mathfrak{n}\) of \(\mathfrak{g}\) containing \(\mathfrak{d} \cdot \mathfrak{g}\) such that \(\mathbb{D} \cdot q_{|\mathfrak{n}}\) is closed in \(\mathfrak{n}^\ast\). We then show that for every \(\mathbb{D}\)-orbit \(\Omega\) in \(g^\ast\) the kernel \(\ker_{C^\ast} (\Omega)\) of \(\Omega\) in the \(C^\ast\)-algebra of \(G\) is \(L^1\)-determined, which means that \(\ker_{C^\ast} (\Omega)\) is the closure of the kernel \(\ker L^1(\Omega )\) of \(\Omega\) in the group algebra \(L^1(G)\). This establishes also a new proof of a result of Ungermann, who obtained the same result for the trivial group \(\mathbb{D} = \text{Ad}(G)\). We finally give an example of a non-closed non-\(\ast\)-regular orbit of an exponential solvable group \(G\) and of a coadjoint orbit \(O \subset \mathfrak{g}^\ast\), for which the corresponding kernel \(\text{ker}_{C^\ast}(\pi_O)\) in \(C^\ast (G)\) is not \(L^1\)-determined.Sobolev-type inequalities for Dunkl operators.https://www.zbmath.org/1456.260192021-04-16T16:22:00+00:00"Velicu, Andrei"https://www.zbmath.org/authors/?q=ai:velicu.andreiThe author studies the Sobolev inequality in the Dunkl setting (as well as other estimates, like Nash inequality or Besov space embeddings), providing a simpler elementary proof of the classical case \(p = 2\) and an extension to the end-point case \(p = 1\), that was previously unknown. The main result shows that if \(N+2\gamma>2\), then there exists \(C>0\) such that
\[
\|f\|_{L^q(\mu_k)}\le C\|\nabla_kf\|_{L^2(\mu_k)},
\]
for all \(f\in C_c^\infty(\mathbb R^N)\), where \(\mu_k\) is the Dunkl measure, \(\nabla_k\) is the Dunkl gradient and \(q=2(N+2\gamma)/(N+2\gamma-2).\) Using some ideas from [\textit {B. Bakry} et al., Indiana Univ. Math. J. 44, No. 4, 1032--1074 (1995; Zbl 0857.26006)], this estimate follows in fact as a consequence of the Nash inequality:
\[
\|f\|^{1+2/(2\gamma+N)}_{L^2(\mu_k)}\le C\|\nabla_kf\|_{L^2(\mu_k)}\|f\|^{2/(2\gamma+N)}_{L^1(\mu_k)}.
\]
A key ingredient in the proofs is the carré-du-champ operator, defined as
\[\Gamma(f)=\frac12(\Delta_k(f^2)-2\Delta_k f).
\]
Further extensions are also considered involving Besov spaces. In the last section, explicit expressions for the best constant are obtained as a consequence of an isoperimetric inequality.
Reviewer: Javier Soria (Madrid)Pizzetti formula on the Grassmannian of 2-planes.https://www.zbmath.org/1456.430052021-04-16T16:22:00+00:00"Eelbode, D."https://www.zbmath.org/authors/?q=ai:eelbode.david"Homma, Y."https://www.zbmath.org/authors/?q=ai:homma.youkow|homma.yuki|homma.yasushi|homma.yuko|homma.yushiThe authors study the Higgs algebras \(H_3\) in the harmonic analysis of the Grassmann manifolds \(\mathrm{Gr}_0(m,2):=\mathrm{SO}(m)/(\mathrm{SO}(m-2)\times \mathrm{SO}(2))\). These algebras appear in the Howe duality where \(\mathrm{SO}(m)\) acts on
\(\mathrm{Gr}_0(m,2)\). In particular, an orthogonal decomposition of the vector space of all poynomials under the joint action of \(\mathrm{SO}(m)\times H_3\) is given. This decomposition is then used to derive a Pizetti formula for integrals of the form \(\int_{\mathrm{Gr}_0(m,2)} f\> d\mu\) with the uniform distribution \(\mu\) on \(\mathrm{Gr}_0(m,2)\). The authors also present a connection to a Pizetti formula for Stiefel manifolds by K.~Coulembier and M.~Kieburg.
Reviewer: Michael Voit (Dortmund)Lie groups of controlled characters of combinatorial Hopf algebras.https://www.zbmath.org/1456.220072021-04-16T16:22:00+00:00"Dahmen, Rafael"https://www.zbmath.org/authors/?q=ai:dahmen.rafael"Schmeding, Alexander"https://www.zbmath.org/authors/?q=ai:schmeding.alexanderA theory of controlled characters of a combinatorial Hopf algebras is introduced, given subgroups of the groups of characters. The model is
the tame Butcher group, seen as a subgroup of the Butcher-Connes-Kreimer group.
A combinatorial Hopf algebra is here a graded connected Hopf algebra, isomorphic to a polynomial algebra, with a particular basis, and the characters take their value in a fixed Banach algebra. A controlled character satisfies a growth condition given by a particular bound. If this bound is compatible with the combinatorial structure of the Hopf algebra, then the set of controlled characters is a subgroup of the group of all characters. It is proved that the group of controlled characters is an infinite-dimensional Lie group and that the underlying group is the Lie algebra of infinitesimal controlled characters. When the Hopf algebra is right-handed, it is shown that the group of controlled characters is regular in Milnor's sense.
Reviewer: Loïc Foissy (Calais)Counting and equidistribution for quaternion algebras.https://www.zbmath.org/1456.111862021-04-16T16:22:00+00:00"Lesesvre, Didier"https://www.zbmath.org/authors/?q=ai:lesesvre.didierSummary: We aim at studying automorphic forms of bounded analytic conductor in the totally definite quaternion algebra setting. We prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure. It leads to answering the Sato-Tate conjectures in this case, and contains the counting law of the universal family, with a power savings error term.The Lind-Lehmer constant for \(\mathbb{Z}_2^r\times\mathbb{Z}_4^s\).https://www.zbmath.org/1456.112042021-04-16T16:22:00+00:00"Mossinghoff, Michael J."https://www.zbmath.org/authors/?q=ai:mossinghoff.michael-j"Pigno, Vincent"https://www.zbmath.org/authors/?q=ai:pigno.vincent"Pinner, Christopher"https://www.zbmath.org/authors/?q=ai:pinner.christopher-gFor a finite abelian group \(G=\mathbb{Z}_{n_{1}}\times \cdots \times \mathbb{Z}_{n_{k}}\), where \(\mathbb{Z}_{n_{j}}\) \((1\leq j\leq k)\) denotes the cyclic group with order \(n_{j}\), define
\[
\lambda (G)=\min \left( \left\{ \prod_{j_{1}=1}^{n_{1}}\dots\prod_{j_{k}=1}^{n_{k}}\left\vert F(e^{i2\pi j_{1}/n_{1}},\dots,e^{i2\pi j_{k}/n_{k}})\right\vert \mid F\in \mathbb{Z}[x_{1},\dots,x_{k}]\right\} \cap \lbrack 2,\infty )\right) .
\]
According to [\textit{D. Lind} et al., Proc. Am. Math. Soc. 133, No. 5, 1411--1416 (2005; Zbl 1056.43005); \textit{D. Desilva} and \textit{C. Pinner}, Proc. Am. Math. Soc. 142, No. 6, 1935--1941 (2014; Zbl 1294.11185)], if \(G\neq \mathbb{Z}_{2}\), then
\[
\lambda (G)\leq \operatorname{card}(G)-1, \tag{*}
\]
\((\lambda (\mathbb{Z}_{p^{n}}),\lambda (\mathbb{Z}_{2^{n}}))=(2,3)\) for any natural number \(n\) and any odd prime \(p\), and (*) is sharp when \(G=\mathbb{Z}_{3}^{n}\), or when \(G=\mathbb{Z}_{2}^{n}\) and \(n\geq 2\).
In the paper under review, the authors continue to investigate the values of \(\lambda (G)\) for \(G\) running through certain families of \(p\)-groups, where \(p\in \{2,3\}\). Mainly, they show that \(\lambda (\mathbb{Z}_{3}\times \mathbb{Z}_{3^{n}})=8\), \(n\geq 3\Rightarrow \lambda (\mathbb{Z}_{2}\times \mathbb{Z}_{2^{n}})=9\), and equality occurs (again) in (*) whenever \(G\neq \mathbb{Z}_{2}\) and the factors of \(G\) are all \(\mathbb{Z}_{2}\) or \(\mathbb{Z}_{4}\).
The proofs of these results are based on a generalization of Lemma 2.1 of the last mentioned reference about a congruence satisfied by the rational integers defining \(\lambda (G)\), when \(G\) is a \(p\)-group.
Reviewer: Toufik Zaïmi (Riyadh)Special functions associated with \(K\)-types of degenerate principal series of \(\mathrm{Sp}(n,\mathbb{C})\).https://www.zbmath.org/1456.220052021-04-16T16:22:00+00:00"Mendousse, Grégory"https://www.zbmath.org/authors/?q=ai:mendousse.gregoryThis article is devoted to the study of special vectors contained in various incarnations of generalized principal series representations induced from maximal parabolic subgroups of the complex symplectic group \(\mathrm{Sp}(n,\mathbf{C})\). After reviewing the well-known decomposition of the Hilbert space \(L^2(S^{4n-1})\) under the natural action of \(\mathrm{Sp}(n)\times\mathrm{Sp}(1)\) in terms of spherical harmonics, the author uses quaternionic geometry to establish the existence and uniqueness of bi-invariant spherical harmonics and determines an explicit hypergeometric equation that they satisfy.
The other main result in the paper concerns certain vectors in the so-called \textit{non-standard} picture of the degenerate principal representations. This picture was introduced by \textit{T. Kobayashi} et al. [J. Funct. Anal. 260, No. 6, 1682--1720 (2011; Zbl 1217.22003)] for real symplectic groups and adapted to the complex case by the reviewer in [J. Funct. Anal. 262, No. 9, 4160--4180 (2012; Zbl 1242.22017)]. It is the image of the the classical non-compact picture under a partial Fourier transform afforded by the fact that the unipotent radicals of the inducing parabolic subgroups are Heisenberg groups. The author calculates the image in this picture of particular highest weight vectors, showing that they can be expressed in terms of modified Bessel functions.
Reviewer: Pierre Clare (Williamsburg)Vector valued polynomials, exponential polynomials and vector valued harmonic analysis.https://www.zbmath.org/1456.430032021-04-16T16:22:00+00:00"Laczkovich, M."https://www.zbmath.org/authors/?q=ai:laczkovich.miklosLet \(G\) be a topological abelian semigroup with unit, \(E\) be a Banach space, and let \(C(G,E)\) stand for the set of all continuous functions from \(G\) into \(E\). A function \(f\in C(G,E)\) is called a generalized polynomial if there is an \(n\ge 0\) such that \(\Delta_{h_1}\dots\Delta_{h_{n+1}} f=0\) for every \(h_1, \dots,h_{n+1}\in G\) in which \(\Delta_h\) denotes the difference operator. A function \(f\in C(G,E)\) is said to be a polynomial if it is a generalized polynomial and the linear span of its translates is of finite dimension; \(f\) is a \(w\)-polynomial if \(u\circ f\) is a polynomial for every \(u\) in the dual space of \(E\), and \(f\) is a local polynomial if it is a polynomial on every finitely generated subsemigroup.
The author proves that each of the classes of polynomials, \(w\)-polynomials, generalized polynomials, local polynomials is contained in the next class. If \(G\) is an abelian group and has a dense subgroup with finite torsion free rank, then these classes coincide. He also introduces the classes of exponential polynomials and \(w\)-exponential polynomials and investigates their representations and connection with polynomials and \(w\)-polynomials. He establishes spectral synthesis and analysis in the class \(C(G,E)\). He shows that, if \(G\) is an infinite and discrete abelian group and \(E\) is a Banach space of infinite dimension, then spectral analysis fails in \(C(G,E)\). If \(G\) is discrete with finite torsion free rank and \(E\) is a Banach space of finite dimension, then he proves that spectral synthesis holds in \(C(G,E)\).
Reviewer: Mohammad Sal Moslehian (Mashhad)A note on the multiple fractional integrals defined on the product of nonhomogeneous measure spaces.https://www.zbmath.org/1456.260092021-04-16T16:22:00+00:00"Kokilashvili, Vakhtang"https://www.zbmath.org/authors/?q=ai:kokilashvili.vakhtang-m"Tsanava, Tsira"https://www.zbmath.org/authors/?q=ai:tsanava.tsiraSummary: In this note we present a trace type inequality in the mixed-norm Lebesgue spaces for multiple fractional integrals defined on an arbitrary measure quasi-metric space.Construction of nonuniform wavelet frames on non-Archimedean fields.https://www.zbmath.org/1456.420362021-04-16T16:22:00+00:00"Ahmad, Owais"https://www.zbmath.org/authors/?q=ai:ahmad.owais"Ahmad, Neyaz"https://www.zbmath.org/authors/?q=ai:ahmad.neyazSummary: A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in \(L^2(\mathbb{R})\) was considered by \textit{J.-P. Gabardo} and \textit{M. Z. Nashed} [J. Funct. Anal. 158, No. 1, 209--241 (1998; Zbl 0910.42018)]. In this setting, the associated translation set \(\Lambda = \{0,r/N\}+2 \mathbb{Z}\) is no longer a discrete subgroup of \(\mathbb{R}\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. The main objective of this paper is to develop oblique and unitary extension principles for the construction nonuniform wavelet frames over non-Archimedean Local fields of positive characteristic. An example and some potential applications are also presented.Regular dilation and Nica-covariant representation on right LCM semigroups.https://www.zbmath.org/1456.430042021-04-16T16:22:00+00:00"Li, Boyu"https://www.zbmath.org/authors/?q=ai:li.boyuGeneralizing the celebrated Sz.~Nagy dilation of a single contraction, Brehmer studied regular dilations back in the early sixties. Since then the notion of regular dilation has been investigated by many researchers and has been generalized to product systems, lattice ordered semigroups, and recently to graph products of \(\mathbb{N}\). It was shown by the author of the present paper in [J. Funct. Anal. 273, No. 2, 799--835 (2017; Zbl 06720587)] that for such graph products, the existence of a \(*\)-regular dilation is equivalent to the existence of a minimal isometric Nica-covariant dilation.
In the paper under review, the author extends this result to right LCM semigroups, which are unital left cancellative semigroups \(P\) such that for any \(p, q\in P\), either \(pP\cap qP=\emptyset\) or \(pP\cap qP=rP\) for some \(r\in P\). Such an element \(r\) can be considered as a right least common multiple of \(p\) and \(q\) (hence the name right LCM). The author also shows the equivalence among a \(*\)-regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result unifies many previous results on regular dilations, including Brehmer's theorem, Frazho-Bunce-Popescu's dilation of noncommutative row contractions, regular dilations on lattice ordered semigroups and graph products of \(\mathbb{N}\). Applications to many important classes of right LCM semigroups are given. In the last section of the paper, the author obtains a characterization of \(*\)-regular representations of graph products of right LCM semigroups and an application to doubly commuting representations of direct sums of right LCM semigroups.
Reviewer: Trieu Le (Toledo)Notes on functions of hyperbolic type.https://www.zbmath.org/1456.430022021-04-16T16:22:00+00:00"Monod, Nicolas"https://www.zbmath.org/authors/?q=ai:monod.nicolasSummary: Functions of hyperbolic type encode representations on real or complex hyperbolic spaces, usually infinite-dimensional. These notes set up the complex case. As applications, we prove the existence of a non-trivial deformation family of representations of \(\mathbf{SU}(1,n)\) and of its infinite-dimensional kin \(\text{Is}(\mathbf{H}_{\mathbf{C}}^{\infty})\). We further classify all the self-representations of \(\text{Is}(\mathbf{H}_{\mathbf{C}}^{\infty})\) that satisfy a compatibility condition for the subgroup \(\text{Is}(\mathbf{H}_{\mathbf{R}}^{\infty})\). It turns out in particular that translation lengths and Cartan arguments determine each other for these representations. In the real case, we revisit earlier results and propose some further constructions.The local sum conjecture in two dimensions.https://www.zbmath.org/1456.111472021-04-16T16:22:00+00:00"Fraser, Robert"https://www.zbmath.org/authors/?q=ai:fraser.robert"Wright, James"https://www.zbmath.org/authors/?q=ai:wright.james-rTranslates of functions on the Heisenberg group and the HRT conjecture.https://www.zbmath.org/1456.420382021-04-16T16:22:00+00:00"Currey, B."https://www.zbmath.org/authors/?q=ai:currey.bradley-n-iii|currey.bradley|currey.brad"Oussa, V."https://www.zbmath.org/authors/?q=ai:oussa.vignon|oussa.vignon-sSummary: We prove that the HRT (Heil, Ramanathan, and Topiwala) Conjecture [\textit{C. Heil} et al., Proc. Am. Math. Soc. 124, No. 9, 2787--2795 (1996; Zbl 0859.42023)] is equivalent to the conjecture that co-central translates of square-integrable functions on the Heisenberg group are linearly independent.Hyperreflexivity constants of the bounded \(n\)-cocycle spaces of group algebras and \(C^*\)-algebras.https://www.zbmath.org/1456.460452021-04-16T16:22:00+00:00"Samei, Ebrahim"https://www.zbmath.org/authors/?q=ai:samei.ebrahim"Farsani, Jafar Soltani"https://www.zbmath.org/authors/?q=ai:farsani.jafar-soltaniSummary: We introduce the concept of strong property \((\mathbb{B})\) with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all \(C^*\)-algebras and group algebras have the strong property \((\mathbb{B})\) with a constant given by \(288 \pi (1 + \sqrt{2})\). We then use this result to find a concrete upper bound for the hyperreflexivity constant of \(\mathcal{Z}^n (A, X)\), the space of bounded \(n\)-cocycles from \(A\) into \(X\), where \(A\) is a \(C^*\)-algebra or the group algebra of a group with an open subgroup of polynomial growth and \(X\) is a Banach \(A\)-bimodule for which \(\mathcal{H}^{n+1} (A, X)\) is a Banach space. As another application, we show that for a locally compact amenable group \(G\) and \(1 < p < \infty\), the space \(CV_P(G)\) of convolution operators on \(L^p(G)\) is hyperreflexive with a constant given by \(384 \pi^2 (1 + \sqrt{2})\). This is the generalization of a well-known result of \textit{E. Christensen} [Math. Scand. 50, 111--122 (1982; Zbl 0503.47032)] for \(p = 2\).Regularity, continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces.https://www.zbmath.org/1456.601222021-04-16T16:22:00+00:00"Cleanthous, Galatia"https://www.zbmath.org/authors/?q=ai:cleanthous.galatia"Georgiadis, Athanasios G."https://www.zbmath.org/authors/?q=ai:georgiadis.athanasios-g"Lang, Annika"https://www.zbmath.org/authors/?q=ai:lang.annika"Porcu, Emilio"https://www.zbmath.org/authors/?q=ai:porcu.emilioSummary: Gaussian random fields defined over compact two-point homogeneous spaces are considered and Sobolev regularity and Hölder continuity are explored through spectral representations. It is shown how spectral properties of the covariance function associated to a given Gaussian random field are crucial to determine such regularities and geometric properties. Furthermore, fast approximations of random fields on compact two-point homogeneous spaces are derived by truncation of the series expansion, and a suitable bound for the error involved in such an approximation is provided.Müntz-Szász type theorems for the density of the span of powers of functions.https://www.zbmath.org/1456.420392021-04-16T16:22:00+00:00"Jaming, Philippe"https://www.zbmath.org/authors/?q=ai:jaming.philippe"Simon, Ilona"https://www.zbmath.org/authors/?q=ai:simon.ilonaSummary: The aim of this paper is to establish density properties in \(L^p\) spaces of the span of powers of functions \(\{ \psi^\lambda : \lambda\in\Lambda\}\), \(\Lambda\subset\mathbb{N}\) in the spirit of the Müntz-Szász Theorem. As density is almost never achieved, we further investigate the density of powers and a modulation of powers \(\{\psi^\lambda,\psi^\lambda e^{i\alpha t}:\lambda\in\Lambda\}\). Finally, we establish a Müntz-Szász Theorem for density of translates of powers of cosines \(\{\cos^\lambda(t-\theta_1),\cos^\lambda(t-\theta_2):\lambda\in\Lambda\}\). Under some arithmetic restrictions on \(\theta_1 - \theta_2\), we show that density is equivalent to a Müntz-Szász condition on \(\Lambda\) and we conjecture that those arithmetic restrictions are not needed. Some links are also established with the recently introduced concept of Heisenberg Uniqueness Pairs.On graph products of multipliers and the Haagerup property for \(C^{\ast}\)-dynamical systems.https://www.zbmath.org/1456.460542021-04-16T16:22:00+00:00"Atkinson, Scott"https://www.zbmath.org/authors/?q=ai:atkinson.scott-eThe following paragraphs are essentially taken from the author's abstract and introduction.
The Haagerup property is an important approximation property for groups and for self-adjoint operator algebras. Since its appearance in Haagerup's seminal article, this property has been the subject of intense study. In 2012, \textit{Z. Dong} and \textit{Z.-J. Ruan} [Integral Equations Oper. Theory 73, No. 3, 431--454 (2012; Zbl 1263.46043)] introduced the Haagerup property for the action of a discrete group \(G\) on a unital \(C^*\)-algebra \(A\).
The author considers the notion of the graph product of actions of discrete groups \(\{G_v\}\) on a \(C^*\)-algebra \(A\) and shows that, under suitable commutativity conditions, the graph product action \(\bigstar_\Gamma \alpha_v:\bigstar_\Gamma G_v\curvearrowright A\) has the Haagerup property if each action \(\alpha_v: G_v\curvearrowright A\) possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, the author introduces the graph product of multipliers associated to the actions and shows that the graph product of positive-definite multipliers is positive definite. These results have impacts on left-transformation groupoids and give an alternative proof of a known result for coarse embeddability. The author also records a cohomological characterization of the Haagerup property for group actions.
Reviewer: Qing Meng (Qufu)