Convolution Algebra Research Articles

$n$-factorization Property of Bilinear Mappings

In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:Xtimes Yto Z$, depended on a natural number $n$ and a cardinal number $kappa$; which is called $n$-factorization property of level $kappa$. Then we study the relation between $n$-factorization property of level $kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity and also strong Arens irregularity. These results may help us to prove some previous problems related to strong Arens irregularity more easier than old. These include some results proved by Neufang in ~cite{neu1} and ~cite{neu}. Some applications to certain bilinear mappings on convolution algebras, on a locally compact group, are also included. Finally, some solutions related to the Ghahramani-Lau conjecture is raised.

Spherical spectral synthesis on hypergroups

We introduce the basic concepts of spherical spectral synthesis on hypergroups, following the ideas of our paper [11]. The commutativity of the hypergroup X is not assumed, we suppose only that (X,K) is a Gelfand pair, that is, K is a compact subhypergroup in X and the convolution algebra of all K-invariant compactly supported complex Borel measures on X is commutative.

On the Positivity of Kirillov’s Character Formula

We give a direct proof for the positivity of Kirillov’s character on the convolution algebra of smooth, compactly supported functions on a connected, simply connected nilpotent Lie group G. Then we use this positivity result to construct a representation of G × G and establish a G × G-equivariant isometric isomorphism between our representation and the Hilbert–Schmidt operators on the underlying representation of G. In fact, we provide a framework in which we establish the positivity of Kirillov’s character for coadjoint orbits of groups such as $\text {SL}(2, \mathbb {R})$ under additional hypotheses that are automatically satisfied in the nilpotent case. These hypotheses include the existence of a real polarization and the Pukanzsky condition.

Finiteness spaces, étale groupoids and their convolution algebras

Given a ring R, we extend Ehrhard’s linearization process by associating to any pre-finiteness space an R-module endowed with a Lefschetz topology. For a semigroup in the category of pre-finiteness spaces, one can endow this R-module with the convolution product to obtain an R-algebra. As examples of pre-finiteness spaces, we study topological spaces with bounded subsets (i.e., included in a compact) taken to be the finitary subsets. We prove that we obtain a finiteness space from any hemicompact space via this construction. As a corollary, any etale Hausdorff groupoid induces a semigroup in pre-finiteness spaces and its associated convolution algebra is complete in the hemicompact case. This is in particular the case for the infinite paths groupoid associated to any countable row-finite directed graph.

Topological centres of weighted convolution algebras

Let G be a non-compact locally compact group with a continuous submultiplicative weight function ω such that ω(e)=1 and ω is diagonally bounded with bound K≥1. When G is σ-compact, we show that ⌊K⌋+1 many points in the spectrum of LUC(ω−1) are enough to determine the topological centre of LUC(ω−1)⁎ and that ⌊K⌋+2 many points in the spectrum of L∞(ω−1) are enough to determine the topological centre of L1(ω)⁎⁎ when G is in addition a SIN-group. We deduce that the topological centre of LUC(ω−1)⁎ is the weighted measure algebra M(ω) and that of C0(ω−1)⊥ is trivial for any locally compact group. The topological centre of L1(ω)⁎⁎ is L1(ω) and that of L0∞(ω)⊥ is trivial for any non-compact locally compact SIN-group. The same techniques apply and lead to similar results when G is a weakly cancellative right cancellative discrete semigroup.

Yang-Baxter algebras, convolution algebras, and Grassmannians

Статья посвящена новому, активно развивающемуся направлению современной математики - изучению связи квантовых интегрируемых моделей и исчисления Шуберта для колчанных многообразий. В статье предлагается геометрическая конструкция решений уравнения Янга-Бакстера и алгебр, связанных с ними, которые называются алгебрами Янга-Бакстера. Эти алгебры играют центральную роль в квантовых интегрируемых системах и точно решаемых (интегрируемых) решеточных моделях статистической физики. Мы покажем на примере классической геометрии многообразий Грассмана, как появляется указанная выше связь. Конкретно, мы отождествляем алгебру конволюций, возникающую в эквивариантном исчислении Шуберта, с алгеброй Янга-Бакстера вырождения асимметричной шестивершинной модели, так называемой пятивершинной модели. Мы покажем также, как, используя наши методы, можно построить действие факторов универсальной обертывающей алгебры для алгебры токов $\mathfrak{sl}_2[t]$ (так называемые алгебры типа Шура) на тензорных произведениях ее представлений вычисления $\mathbb{C}^2[t]$. Наконец, мы связываем нашу конструкцию с когомологической алгеброй Холла для колчана $A_1$. Библиография: 125 названий.

A theorem for random Fourier series on compact quantum groups

Helgason showed that a given measure $f\in M(G)$ on a compact group $G$ should be in $L^2(G)$ automatically if all random Fourier series of $f$ are in $M(G)$. We explore a natural analogue of the theorem in the framework of compact quantum groups and apply the obtained results to study complete representability problem for convolution algebras of compact quantum groups as an operator algebra.

On the size of orbits in the duals of 𝐶*-algebras and convolution algebras

We solve an open problem raised in [Trans. Amer. Math. Soc. 366 (2014), pp.4151–4171] concerning (infinite-dimensional) commutative semisimple Banach algebras A \mathcal {A} with a bounded approximate identity, namely, whether there always exists a functional f ∈ A ∗ f \in \mathcal {A}^* such that the orbit subspace A ∗ ∗ ◻ f \mathcal {A}^{**} \Box f of A ∗ \mathcal {A}^* is w ∗ w^* -closed and infinite-dimensional. Indeed, on the one hand, we show that no commutative C ∗ C^* -algebra shares this property. On the other hand, we prove that the answer is positive, in a strong sense, in the case of convolution algebras A = L 1 ( G ) \mathcal {A} = L_1(\mathcal {G}) , for large classes of locally compact groups G \mathcal {G} (commutativity is not needed): there exists f ∈ A ∗ f \in \mathcal {A}^* with maximal orbit, in fact, f f satisfies B a l l ( A ∗ ) = B a l l ( A ∗ ∗ ) ◻ f \mathrm {Ball} (\mathcal {A}^*) = \mathrm {Ball}(\mathcal {A}^{**}) \Box f . Moreover, as we shall see, the latter property links the size of orbits to Arens irregularity.

Renewal theory for extremal Markov sequences of Kendall type

The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions we prove a Blackwell theorem and a limit theorem for renewal processes defined by Kendall random walks.Our results set new research hypotheses for other generalized convolution algebras to investigate renewal processes constructed by Markov processes with respect to generalized convolutions.

Topological tensor product of bimodules, complete Hopf algebroids and convolution algebras

Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.

Norm-Controlled Inversion in Weighted Convolution Algebras

Let G be a discrete group, let $$p\ge 1$$, and let $$\omega $$ be a weight on G. Using the approach from Gröchenig and Klotz (J Lond Math Soc (2) 88:49–64, 2013), we provide sufficient conditions on a weight $$\omega $$ for $$\ell ^p(G,\omega )$$ to be a Banach algebra admitting a norm-controlled inversion in $$C^*_r(G)$$. We show that our results can be applied to various cases including locally finite groups as well as finitely generated groups of polynomial or intermediate growth and a natural class of weights on them. These weights are of the form of polynomial or certain subexponential functions. We also consider the non-discrete case and study the existence of norm-controlled inversion in $$B(L^2(G))$$ for some related convolution algebras.

Peter–Weyl Iwahori Algebras

AbstractThe Peter–Weyl idempotent $e_{\mathscr{P}}$ of a parahoric subgroup $\mathscr{P}$ is the sum of the idempotents of irreducible representations of $\mathscr{P}$ that have a nonzero Iwahori fixed vector. The convolution algebra associated with $e_{\mathscr{P}}$ is called a Peter–Weyl Iwahori algebra. We show that any Peter–Weyl Iwahori algebra is Morita equivalent to the Iwahori–Hecke algebra. Both the Iwahori–Hecke algebra and a Peter–Weyl Iwahori algebra have a natural conjugate linear anti-involution $\star$, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by $\bullet$, and the Morita equivalence preserves irreducible and unitary modules for $\bullet$.

Beyond Wiener’s Lemma: Nuclear Convolution Algebras and the Inversion of Digital Filters

A convolution algebra is a topological vector space $\mathcal{X}$ that is closed under the convolution operation. It is said to be inverse-closed if each element of $\mathcal{X}$ whose spectrum is bounded away from zero has a convolution inverse that is also part of the algebra. The theory of discrete Banach convolution algebras is well established with a complete characterization of the weighted $\ell_1$ algebras that are inverse-closed and referred to as the Gelfand-Raikov-Shilov (GRS) spaces. Our starting point here is the observation that the space $\mathcal{S}(\mathbb{Z}^d)$ of rapidly decreasing sequences, {which is not Banach but nuclear}, is an inverse-closed convolution algebra. This property propagates to the more constrained space of exponentially decreasing sequences $\mathcal{E}(\mathbb{Z}^d)$ that we prove to be nuclear as well. Using a recent extended version of the GRS condition, we then show that $\mathcal{E}(\mathbb{Z}^d)$ is actually the smallest inverse-closed convolution algebra. This allows us to describe the hierarchy of the inverse-closed convolution algebras from the smallest, $\mathcal{E}(\mathbb{Z}^d)$, to the largest, $\ell_{1}(\mathbb{Z}^d)$. In addition, we prove that, in contrast to $\mathcal{S}(\mathbb{Z}^d)$, all members of $\mathcal{E}(\mathbb{Z}^d)$ admit well-defined convolution inverses in $\mathcal{S}'(\mathbb{Z}^d)$ with the "unstable" scenario (when some frequencies are vanishing) giving rise to inverse filters with slowly-increasing impulse responses. Finally, we use those results to reveal the decay and reproduction properties of an extended family of cardinal spline interpolants.

Algebraic Hopf invariants and rational models for mapping spaces

The main goal of this paper is to define an invariant $$mc_{\infty }(f)$$ of homotopy classes of maps $$f:X \rightarrow Y_{\mathbb {Q}}$$ , from a finite CW-complex X to a rational space $$Y_{\mathbb {Q}}$$ . We prove that this invariant is complete, i.e. $$mc_{\infty }(f)=mc_{\infty }(g)$$ if and only if f and g are homotopic. To construct this invariant we also construct a homotopy Lie algebra structure on certain convolution algebras. More precisely, given an operadic twisting morphism from a cooperad $$\mathcal {C}$$ to an operad $$\mathcal {P}$$ , a $$\mathcal {C}$$ -coalgebra C and a $$\mathcal {P}$$ -algebra A, then there exists a natural homotopy Lie algebra structure on $$Hom_\mathbb {K}(C,A)$$ , the set of linear maps from C to A. We prove some of the basic properties of this convolution homotopy Lie algebra and use it to construct the algebraic Hopf invariants. This convolution homotopy Lie algebra also has the property that it can be used to model mapping spaces. More precisely, suppose that C is a $$C_\infty $$ -coalgebra model for a simply-connected finite CW-complex X and A an $$L_\infty $$ -algebra model for a simply-connected rational space $$Y_{\mathbb {Q}}$$ of finite $$\mathbb {Q}$$ -type, then $$Hom_\mathbb {K}(C,A)$$ , the space of linear maps from C to A, can be equipped with an $$L_\infty $$ -structure such that it becomes a rational model for the based mapping space $$Map_*(X,Y_\mathbb {Q})$$ .

Convolution algebras and the deformation theory of infinity-morphisms

Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for infinity-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and infinity-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts infinity-morphisms in both slots and which is well defined up to homotopy, and we generalize and take a new point of view on some other already known results. This paper concludes a series of works by the two authors dealing with the investigation of convolution algebras.

Weighted Khovanov-Lauda-Rouquier Algebras

In this paper, we define a generalization of Khovanov–Lauda–Rouquier algebras which we call weighted Khovanov–Lauda–Rouquier algebras . We show that these algebras carry many of the same structures as the original Khovanov–Lauda–Rouquier algebras, including induction and restriction functors which induce a twisted bialgebra structure on their Grothendieck groups. We also define natural steadied quotients of these algebras, which in an important special cases give categorical actions of an associated Lie algebra. These include the algebras categorifying tensor products and Fock spaces defined by the author and C. Stroppel [ B. Webster , Mem. Am. Math. Soc. 1191, iii-vi, 146 p. (2017; Zbl 07000045), p. 141, and C. Stroppel and B. Webster , “Quiver Schur algebras and q -Fock space”, Preprint, ]. For symmetric Cartan matrices, weighted KLR algebras also have a natural geometric interpretation as convolution algebras, generalizing that for the original KLR algebras by M. Varagnolo and E. Vasserot [J. Reine Angew. Math. 659, 67–100 (2011; Zbl 1229.17019)]; this result has positivity consequences important in the theory of crystal bases. In this case, we can also relate the Grothendieck group and its bialgebra structure to the Hall algebra of the associated quiver.

Orthogonally additive polynomials on convolution algebras associated with a compact group

Let $G$ be a compact group, let $X$ be a Banach space, and let $P\colon L^1(G)\to X$ be an orthogonally additive, continuous $n$-homogeneous polynomial. Then we show that there exists a unique continuous linear map $\Phi\colon L^1(G)\to X$ such that $P(f)=\Phi \bigl(f\ast\stackrel{n}{\cdots}\ast f \bigr)$ for each $f\in L^1(G)$. We also seek analogues of this result about $L^1(G)$ for various other convolution algebras, including $L^p(G)$, for $1< p\le\infty$, and $C(G)$.

On the similarity problem for locally compact quantum groups

A well-known theorem of Day and Dixmier states that any uniformly bounded representation of an amenable locally compact group G on a Hilbert space is similar to a unitary representation. Within the category of locally compact quantum groups, the conjectural analogue of the Day–Dixmier theorem is that every completely bounded Hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a ⁎-representation. We prove that this conjecture is false for a large class of non-Kac type compact quantum groups, including all q-deformations of compact simply connected semisimple Lie groups. On the other hand, within the Kac framework, we prove that the Day–Dixmier theorem does indeed hold for several new classes of examples, including amenable discrete quantum groups of Kac-type.

Orlicz Algebras on Homogeneous Spaces of Compact Groups and Their Abstract Linear Representations

Let G be a compact group, H a closed subgroup of G and let m be the normalized G-invariant measure on the homogeneous space G / H obtained from Weil’s formula. In this article, for a given Young function $$\varphi $$ , we give a new class of Banach convolution algebras on homogeneous spaces of compact groups by introducing a convolution and an involution on the Orlicz space $$L^\varphi (G/H, m)$$ . Finally, a class of linear representations of this class of Banach convolution algebras is presented.

Jordan, Jordan Right and Jordan Left Derivations on Convolution Algebras

In this paper, we investigate Jordan derivations, Jordan right derivations and Jordan left derivations of $$L_0^\infty ({{\mathcal {G}}})^*$$ . We show that any Jordan (right) derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ is a (right) derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ and the zero map is the only Jordan left derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ . Then, we prove that the range of a Jordan (right) derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ is contained into $$\hbox {rad}(L_0^\infty ({{\mathcal {G}}})^*)$$ . Finally, we establish that the product of two Jordan (right) derivations of $$L_0^\infty ({{\mathcal {G}}})^*$$ is always a derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ and there is no nonzero centralizing Jordan (right) derivation on $$L_0^\infty ({{\mathcal {G}}})^*$$ .