Convolution Algebra Research Articles

On one-sided strong Arens irregularity

We present a natural class of Banach algebras which are one-sided strongly Arens irregular. More precisely, for any non-compact, second countable locally compact group $${\mathcal{G}}$$ , we show that the convolution algebras $${\mathcal{N}}(L_{p}({\mathcal{G}}))$$ of nuclear operators on $$L_{p}({\mathcal{G}}), p \in (1,\infty)$$ , as introduced and studied by the author in [3], [6], are left but not right strongly Arens irregular.

Weakly almost periodic functionals on the measure algebra

It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C*-algebra. This implies that the weakly almost periodic functionals on M(G), the measure algebra of a locally compact group G, is a C*-subalgebra of M(G)* = C 0(G)**. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps. The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces.

Bass and Topological Stable Ranks of Complex and Real Algebras of Measures, Functions and Sequences

We compute the Bass stable rank and the topological stable rank of several convolution Banach algebras of complex measures on (-∞,∞) or on [0,∞) consisting of a discrete measure and/or of an absolutely continuous measure. We also compute the stable ranks of the convolution algebras , , l1(S) and , where S is an arbitrary subgroup of , of the almost periodic algebra AP and of , etc. We answer affirmatively the question posed by Mortini (Studia Mathematica 103(3):275–281, 1992). For the above algebras, the polydisc algebra , the algebra of continuous functions, and others, we also study their subsets (real Banach algebras) of real-valued measures, real-valued sequences or real-symmetric functions, and of corresponding exponentially stable algebras (for example, the Callier–Desoer algebra of causal exponentially decaying measures and L1 functions), and we compute their stable ranks. Finally, we show that in some of these real algebras a variant of the parity interlacing property is equivalent to reducibility of a unimodular (or coprime) pair. Also corona theorems are presented and the existence of coprime fractions is studied; in particular, we list which of these algebras are Bezout domains.

Weighted convolution algebras on subsemigroups of the real line

Weighted convolution algebras on subsemigroups of the real line

Representation Theory of Finite Dimensional Algebras

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

HOCHSCHILD HOMOLOGY AND COHOMOLOGY OF 1(ZFormula)

Building on the recent determination of the simplicial cohomology groups of the convolution algebra ${\ell}^1({\mathbb Z}_+^k)$ [Gourdeau, Lykova, White, 2005] we investigate what can be said for cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of Harrison homology and cohomology in the context of Banach algebras, and a development of some of its basic features. As an application of our techniques we reprove some known results on second-degree cohomology.

Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ℝ,ℂ,ℍ were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.

Arens multiplication on Banach algebras related to locally compact semigroups

AbstractLet S be a locally compact semigroup, let ω be a weight function on S, and let Ma (S, ω) be the weighted semigroup algebra of S. Let L∞0 (S;Ma (S, ω)) be the C*‐algebra of allMa (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L∞0 (S;Ma (S, ω))* under which Ma (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L∞0 (S;Ma (S, ω))*. We then study Arens regularity of L∞0 (S;Ma (S, ω))* and ist relation with Arens regularity of Ma (S, ω), M (S, ω) and the discrete convolution algebra ℓ1(S, ω). As the main result, we prove that L∞0 (S;Ma (S, ω))* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Coverings, correspondences, and noncommutative geometry

We construct an additive category where objects are embedded graphs in the 3-sphere and morphisms are geometric correspondences given by 3-manifolds realized in different ways as branched covers of the 3-sphere, up to branched cover cobordisms. We consider dynamical systems obtained from associated convolution algebras endowed with time evolutions defined in terms of the underlying geometries. We describe the relevance of our construction to the problem of spectral correspondences in noncommutative geometry.

Algebra homomorphisms from cosine convolution algebras

In this paper we deal with the weighted Banach algebra L 1 (ℝ+, * c ), where *c is the cosine convolution product. We describe its character space and its multiplier algebra. Our main results concern bounded algebra homomorphisms from L 1 (ℝ+, * c ). We give a variant of Kisynski’s theorem for such homomorphisms and characterize them in terms of integrated cosine functions. A generalized form of the Sova-Da Prato-Giusti theorem about generation of cosine functions is also given.

The equivariant index theorem in entire cyclic cohomology

AbstractLet G be a locally compact group acting smoothly and properly by isometries on a complete Riemannian manifold M, with compact quotient G\M. There is an assembly map which associates to any G-equivariant K-homology class on M, an element of the topological K-theory of a suitable Banach completion of the convolution algebra of continuous compactly supported functions on G. The aim of this paper is to calculate the composition of the assembly map with the Chern character in entire cyclic homology . We prove an index theorem reducing this computation to a cup-product in bivariant entire cyclic cohomology. As a consequence we obtain an explicit localization formula which includes, as particular cases, the equivariant Atiyah-Segal-Singer index theorem when G is compact, and the Connes-Moscovici index theorem for G-coverings when G is discrete. The proof is based on the bivariant Chern character introduced in previous papers.

Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids

Any étale Lie groupoid G is completely determined by its associated convolution algebra C c ∞( G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C c ∞( G)- C c ∞( H)-bimodule C c ∞( P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C c ∞gives an equivalence between the Morita category of étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.

Bornological quantum groups

We introduce and study the concept of a bornological quantum group. This generalizes the theory of algebraic quantum groups in the sense of van Daele from the algebraic setting to the framework of bornological vector spaces. Working with bornological vector spaces, the scope of the latter theory can be extended considerably. In particular, the bornological theory covers smooth convolution algebras of arbitrary locally compact groups and their duals. Moreover Schwartz algebras of nilpotent Lie groups are bornological quantum groups in a natural way, and similarly one may consider algebras of functions on finitely generated discrete groups defined by various decay conditions. Another source of examples arises from deformation quantization in the sense of Rieffel. Apart from describing these examples we obtain some general results on bornological quantum groups. In particular, we construct the dual of a bornological quantum group and prove the Pontrjagin duality theorem.

Some algebraic relations between involutions, convolutions and correlations, with applications to holographic memories

Convolutions and correlations # in spaces H of doubly infinite sequences are related by a # b = S(a Sb), where S is an involution which reflects the order in the integral domain Z on which the sequences are defined. This relation can be used to represent a non-associative correlation algebra (H, #) by an associative convolution algebra equipped with the involution S which, as is shown, greatly simplifies derivations. Related matrix representations of #, S are given for sequences with finite support in Ren. Some implications for holographic memory models are discussed.

On conformal bialgebras

We study conformal algebras from the point of view of conformal dual of classical Lie coalgebra structures. We define the notions of Lie conformal coalgebra and bialgebra. We obtain a conformal analog of the CYBE, the Manin triples and Drinfeld's double. With the definition of vertex duals, we obtain a natural description of the Lie algebra associated to a conformal algebra as a convolution algebra, clarifying the classical constructions in the theory of conformal algebras and vertex algebras.

On the topological centre problem for weighted convolution algebras and semigroup compactifications

Let G be a locally compact, non-compact group (we make the non-compactness assumption, for the most part, simply to avoid trivialities). We show that under a very mild assumption on the weight function w, the weighted group algebra L 1 (G,w) is strongly Arens irregular in the sense of Dales and Lau; i.e., both topological centres of L 1 (G, w)** equal L 1 (G, w). Also, we show that the topological centre of the algebra LUC (G,w -1 )* equals the weighted measure algebra M(G, w). Moreover, still in the same situation, we prove that every linear (left) L ∞ (G, w)*-module map on L ∞ (G, w -1 ) is automatically bounded, and even w*-w*-continuous, hence given by convolution with an element in M(G, w). To this end, we derive a general factorization theorem for bounded families in the L ∞ (G,w -1 )*-module L ∞ (G,w -1 ). Finally, using this result in the case where w ≡ 1, we give a short proof of a theorem due to Protasov and Pym, stating that the topological centre of the semigroup G LUC G is empty, where G LUC denotes the LUC-compactification of G. This sharpens an earlier result by Lau and Pym; moreover, our method of proof gives a partial answer to a problem raised by Lau and Pym in 1995.

An unpublished theorem of Manfred Schocker and the Patras-Reutenauer algebra

Patras, Reutenauer (J. Algebr. Comb. 16:301–314, [2002]) describe a subalgebra \(\mathfrak{A}\) of the Malvenuto-Reutenauer algebra ℘. Their paper contains several characteristic properties of this subalgebra. In an unpublished manuscript Manfred Schocker states without proof a theorem, providing two further characterizations of the Patras-Reutenauer algebra. In this paper we establish a slightly generalized version of Schocker’s theorem, and give some applications. Finally we describe a derivation of the convolution algebra \(\mathfrak{A}\) , which is a homomorphism for the inner product.

Banach algebras of pseudodifferential operators and their almost diagonalization

We define new symbol classes for pseudodifferential operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra 𝒜 over a lattice Λ we associate a symbol class M ∞,𝒜 . Then every operator with a symbol in M ∞,𝒜 is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra 𝒜. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on L 2 (ℝ d ). If a version of Wiener’s lemma holds for 𝒜, then the algebra of pseudodifferential operators is closed under inversion. The theory contains as a special case the fundamental results about Sjöstrand’s class and yields a new proof of a theorem of Beals about the Hörmander class S 0,0 0 .

Biflatness of ℓ1-Semilattice Algebras

We show that if L is a semilattice then the l1-convolution algebra of L is biflat precisely when L is uniformly locally finite. Our proof technique shows in passing that if this convolution algebra is biflat then it is isomorphic as a Banach algebra to the Banach space l1(L) equipped with pointwise multiplication. At the end we sketch how these techniques may be extended to prove an analogous characterisation of biflatness for Clifford semigroup algebras.

Explicit study of lie group convolution algebras as deformation quantizations

Lie group convolution algebras are regarded as deformation quantizations of linear Poisson brackets. It is shown that this deformation is a star product equivalent to the Kontsevich star product, some interesting properties of this star product are proved, and an explicit expression for the star product is given.