Convolution Algebra Research Articles

Slowly oscillating functions in semigroup compactifications and convolution algebras

The paper proposes a unified approach to many key theorems proved in the last twenty years in different areas of abstract harmonic analysis. This approach is based on the so-called slowly oscillating functions which were introduced in coarse geometry. In addition to this method being the most natural and simple, it also leads to the generalisation of some of the results and to the achievement of some new results. Several of these results concern the topological centres of convolution algebras and semigroup compactifications.

Bessel convolutions on matrix cones

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras ${\mathbb F} = {\mathbb R}, {\mathbb C}$ or ${\mathbb H}$ which interpolate the convolution algebras of radial bounded Borel measures on a matrix space $M_{p,q}({\mathbb F})$ with $p\geq q$. Radiality in this context means invariance under the action of the unitary group $U_p({\mathbb F})$ from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank-one case, namely the Bessel–Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper we study structures depending only on the matrix spectra. Under the mapping $r\mapsto \text{spec}(r)$, the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type $B_q$. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds $U(p,q)/(U_p\times U_q)$ over ${\mathbb F}$.

Simplicial Cohomology of Some Semigroup Algebras

AbstractIn this paper, we investigate the higher simplicial cohomology groups of the convolution algebra ℓ1(S) for various semigroups S. The classes of semigroups considered are semilattices, Clifford semigroups, regular Rees semigroups and the additive semigroups of integers greater than a for some integer a. Our results are of two types: in some cases, we show that some cohomology groups are 0, while in some other cases, we show that some cohomology groups are Banach spaces.

On the topological centre of the algebra [formula omitted] for general topological groups

We consider the Banach algebra LUC ( G ) ∗ for a not necessarily locally compact topological group G . Our goal is to characterize the topological centre Z t ( LUC ( G ) ∗ ) of LUC ( G ) ∗ . For locally compact groups G , it is well known that Z t ( LUC ( G ) ∗ ) equals the measure algebra M ( G ) . We shall prove that for every second countable (not precompact) group G , we have Z t ( LUC ( G ) ∗ ) = M ( G ˆ ) , where G ˆ denotes the completion of G with respect to its right uniform structure (if G is precompact, then Z t ( LUC ( G ) ∗ ) = LUC ( G ) ∗ , of course). In fact, this will follow from our more general result stating that for any separable (or any precompact) group G , we have Z t ( LUC ( G ) ∗ ) = Leb ( G ) , where Leb ( G ) denotes the algebra of uniform measures. The latter result also partially answers a conjecture made by I. Csiszár 35 years ago [I. Csiszár, On the weak ∗ continuity of convolution in a convolution algebra over an arbitrary topological group, Studia Sci. Math. Hungar. 6 (1971) 27–40]. We shall give similar results for the topological centre Λ ( G LUC ) of the LUC-compactification G LUC of G . In particular, we shall prove that for any second countable (not precompact) group G admitting a group completion, we have Λ ( G LUC ) = G ˆ (if G is precompact, then Λ ( G LUC ) = G LUC ). Finally, we shall show that every linear (left) LUC ( G ) ∗ -module map on LUC ( G ) is automatically continuous whenever G is, e.g., separable and not precompact.

Wave front set for positive operators and for positive elements in non-commutative convolution algebras

Wave front set for positive operators and for positive elements in non-commutative convolution algebras

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M , the groupoid convolution algebra C c ∞ ( G ) of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra C c ∞ ( M ) which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over C c ∞ ( M ) we construct the associated spectral étale Lie groupoid G sp ( A ) over M such that G sp ( C c ∞ ( G ) ) is naturally isomorphic to G . Both these constructions are functorial, and C c ∞ is fully faithful left adjoint to G sp . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid C c ∞ ( G ) of an étale Lie groupoid G .

Homomorphisms of the algebra of locally integrable functions on the half line

AbstractLet φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.

Nonstandard Ideals from Nonstandard Dual Pairs for L1(ω) and l1(ω)

AbstractThe Banach convolution algebras l1(ω) and their continuous counterparts L1(ℝ+, ω) are much studied, because (when the submultiplicative weight function ω is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of “nice” weights ω, the only closed ideals they have are the obvious, or “standard”, ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in l1(ω). His proof was successfully exported to the continuous case L1(ℝ+, ω) by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in l1(ω) and L1(ℝ+, ω). The new proof is based on the idea of a “nonstandard dual pair” which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in L1(ℝ+, ω) containing functions whose supports extend all the way down to zero in ℝ+, thereby solving what has become a notorious problem in the area.

Weighted L p -algebras on groups

The space Lp(G), 1 > p < ∞, on a locally compact group G is known to be closed under convolution only if G is compact. However, the weighted spaces Lp(G, w) are Banach algebras with respect to convolution and natural norm under certain conditions on the weight. In the present paper, sufficient conditions for a weight defining a convolution algebra are stated in general form. These conditions are well known in some special cases. The spectrum (the maximal ideal space) of the algebra Lp(G,w) on an Abelian group G is described. It is shown that all algebras of this type are semisimple.

SIMPLICIAL HOMOLOGY AND HOCHSCHILD COHOMOLOGY OF BANACH SEMILATTICE ALGEBRAS

The ${\ell}^1$-convolution algebra of a semilattice is known to have trivial cohom ology in degrees 1,2 and 3 whenever the coefficient bimodule is symmetric. We ex tend this result to all cohomology groups of degree $\geq 1$ with symmetric coef ficients. Our techniques prove a stronger splitting result, namely that the spli tting can be made natural with respect to the underlying semilattice.

Convolution over Lie and Jordan algebras

Given a ternary relation C on a set U and an algebra A, we present a construction of a convolution algebra A(U, C) of U = (U, C) over A. This generalises bothmatrix algebras and algebras obtained from convolution of monoids. To any class of algebras corresponds a class of convolution structures. Our study cases are the classes of commutative, associative, Lie, and Jordan algebras. In each of these classes we give conditions on (U, C) under which A(U, C) is in the same class as A. It turns out that in some situations these conditions are even necessary.

Homology of formal deformations of proper étale Lie groupoids

In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a non-commutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all traces on this formal deformation, and provide an explicit construction.

Time frequency analysis and multipliers of the spaces $M(p,q) (R^d)$ and $S(p,q) (R^d)$

In the second section of this paper, in analogy to modulation spaces, we define the space $M(p, q)(R^{d})$ to be the subspace of tempered distributions $f \in S'(R^{d})$ such that the Gabor transform $V_{g}(f)$ of $f$ is in the Lorentz space $L(p, q)(R^{2d})$, where the window function $g$ is a rapidly decreasing function. We endow this space with a suitable norm and show that the $M(p, q)(R^{d})$ becomes a Banach space and is invariant under time-frequency shifts for $1 \leq p,q \leq \infty$. We also discuss the dual space of $M(p, q)(R^{d})$ and the multipliers from $L^{1}(R^{d})$ into $M(p, q)(R^{d})$. In the third section we intend to study the intersection space $S (p, q)(R^{d}) = L^{1}(R^{d})\cap M (p, q)(R^{d})$ for $1 < p < \infty$, $1 \leq q \leq \infty$. We endow it with the sum norm and show that $S (p, q)(R^{d})$ becomes a Banach convolution algebra. Further we prove that it is also a Segal algebra. In the last section we discuss the multipliers of $S (p, q)(R^{d})$ and $M (p, q)(R^{d})$.

Positivity in twisted convolution algebra and Fourier modulation spaces

Let Wp,q be the Fourier modulation space FMp,q and let *? be the twisted convolution. I? ? ? D' such that (a *? ?,?)? 0 for every ? ? C?0, and ? ? such that X(0) ? 0, then we prove that ?? ? Wp,? iff ? ? Wp,?. We also present some extensions to the case when weighted Fourier modulation spaces are used. AMS Mathematics Subject Classification (2000): 47B65, 35A21, 35S05.

Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces

Consider a compact locally symmetric space M of rank r , with fundamental group Γ . The von Neumann algebra VN ( Γ ) is the convolution algebra of functions f ∈ ℓ 2 ( Γ ) which act by left convolution on ℓ 2 ( Γ ) . Let T r be a totally geodesic flat torus of dimension r in M and let Γ 0 ≅ Z r be the image of the fundamental group of T r in Γ . Then VN ( Γ 0 ) is a maximal abelian ★ -subalgebra of VN ( Γ ) and its unitary normalizer is as small as possible. If M has constant negative curvature then the Pukánszky invariant of VN ( Γ 0 ) is ∞ .

Asymptotic expansions of convolutions of regularly varying distributions

AbstractIn this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails.Higher-order asymptotics are also obtained when considering asemiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task.

A singular measure on the Cantor group

Let Ω = { − 1 , 1 } N and { ω j } be independent random variables taking values in { − 1 , 1 } with equal probability. Endowed with the product topology and under the operation of pointwise product, Ω is a compact Abelian group, the so-called Cantor group. Let a , b , c be real numbers with 1 + a + b + c > 0 , 1 + a − b − c > 0 , 1 − a + b − c > 0 and 1 − a − b + c > 0 . Finite products on Ω, P n = ∏ j = 1 n ( 1 + a ω j + b ω j + 1 + c ω j ω j + 1 ) , are studied. We show that the weak limit of { P n d ω ∫ Ω P n d ω } exists in the topology of M ( Ω ) , where M ( Ω ) is the convolution algebra of all Radon measure on Ω, thus defined a probability measure on Ω. We also prove that the measure is continuous and singular with respect to the normalized Haar measure on Ω.

Multiplicity One for Representations Corresponding to Spherical Distribution Vectors of Class ?

In this paper one considers a unimodular second countable locally compact group $G$ and the homogeneous space $X:=H /G$, where $H$ is a closed unimodular subgroup of $G$. Over $X$ complex vector bundles are considered such that $H$ acts on the fibers by a unitary representation $\rho$ with closed image. The natural action of $G$ on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class $\rho$. The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that $H$ is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for $X$ a symmetric $k$-variety $\mathcal{H}_k / \mathcal{G}_k$, with $k$ a locally compact field of characteristic not equal to two. Here $\mathcal{G}$ is a reductive algebraic group defined over $k$ and $\mathcal{H}$ is the fixed point group of an involution $\sigma$ of $\mathcal{G}$ defined over $k$. It is shown then that the natural representation $\mathcal{L}$ of $G_k$ on the Hilbert space $L^2(\mathcal{H}_k /\mathcal{G}_k)$ is multiplicity free if $\mathcal{H}$ is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the nonarchimedean case, a general procedure is given that might lead to showing that a pair $(\mathcal{G}_k,\mathcal{H}_k)$ is a generalized Gelfand pair. Here $\mathcal{G}$ and $\mathcal{H}$ are suitable algebraic groups defined over $k$.

Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators

AbstractFor a locally compact group G, the convolution product on the space 𝒩(Lp(G)) of nuclear operators was defined by Neufang [11]. We study homological properties of the convolution algebra 𝒩(Lp(G)) and relate them to some properties of the group G, such as compactness, finiteness, discreteness, and amenability.

Convolution number evaluation of wave-making velocity

Comparison between Gauss quadrature and quasi-analytical integration by Taylorseries expansions using convolution algebra, is made of some integrals that occur in the wave-making vortex integrals. Convolution number integration takes longer, but it permits the evaluation of the accuracy of numerical integration.