Convolution Algebra Research Articles

Constructing non-AMNM weighted convolution algebras for every semilattice of infinite breadth

The AMNM property for commutative Banach algebras is a form of Ulam stability for multiplicative linear functionals. We show that on any semilattice of infinite breadth, one may construct a weight for which the resulting weighted convolution algebra fails to have the AMNM property. Our work is the culmination of a trilogy started in [4] and continued in [5]. In particular, we obtain a refinement of the main result of [5], by establishing a dichotomy for union-closed set systems that has a Ramsey-theoretic flavour.

Convolution Algebras of Double Groupoids and Strict 2-Groups

Double groupoids are a type of higher groupoid structure that can arise when one has two distinct groupoid products on the same set of arrows. A particularly important example of such structures is the irrational torus and, more generally, strict 2-groups. Groupoid structures give rise to convolution operations on the space of arrows. Therefore, a double groupoid comes equipped with two product operations on the space of functions. In this article we investigate in what sense these two convolution operations are compatible. We use the representation theory of compact Lie groups to get insight into a certain class of 2-groups.

Gauge theory on ρ-Minkowski space-time

We construct a gauge theory model on the 4-dimensional ρ-Minkowski space-time, a particular deformation of the Minkowski space-time recently considered. The corresponding star product results from a combination of Weyl quantization map and properties of the convolution algebra of the special Euclidean group. We use noncommutative differential calculi based on twisted derivations together with a twisted notion of noncommutative connection. The twisted derivations pertain to the Hopf algebra of ρ-deformed translations, a Hopf subalgebra of the ρ-deformed Poincaré algebra which can be viewed as defining the quantum symmetries of the ρ-Minkowski space-time. The gauge theory model is left invariant under the action of the ρ-deformed Poincaré algebra. The kinetic part of the action is found to coincide with the one of the usual (commutative) electrodynamics.

Hopf algebroids and twists for quantum projective spaces

We study the relationship between antipodes on a Hopf algebroid ▪ in the sense of Böhm–Szlachanyi and the group of twists that lies inside the associated convolution algebra. We specialize to the case of a faithfully flat H-Hopf–Galois extensions B⊆A and related Ehresmann–Schauenburg bialgebroid. In particular, we find that the twists are in one-to-one correspondence with H-comodule algebra automorphism of A. We work out in detail the U(1)-extension O(CPqn−1)⊆O(Sq2n−1) on the quantum projective space and show how to get an antipode on the bialgebroid out of the K-theory of the base algebra O(CPqn−1).

Ideals in the Convolution Algebra of Periodic Distributions

The ring of periodic distributions on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{\ exttt {d}}$$\\end{document} with usual addition of distributions, and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring S′(Zd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}'(\\mathbb {Z}^{\ exttt {d}})$$\\end{document} of all maps f:Zd→C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {Z}^{\ exttt {d}}\\rightarrow \\mathbb {C}$$\\end{document} of at most polynomial growth (that is, there exist a real number M>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M>0$$\\end{document} and an integer m≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ exttt {m}\\ge 0$$\\end{document} such that |f(n)|≤M(1+|n1|+⋯+|nd|)m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ |f(\\varvec{n})|\\le M(1+|\ exttt{n}_1|+\\cdots +|\ exttt {n}_{\ exttt {d}}|)^{\ exttt {m}}$$\\end{document} for all n=(n1,⋯,nd)∈Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{n}=(\ exttt{n}_1,\\cdots , \ exttt {n}_{\ exttt {d}})\\in \\mathbb {Z}^{\ exttt {d}}$$\\end{document}), with pointwise operations. It is shown that finitely generated ideals in S′(Zd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}'(\\mathbb {Z}^{\ exttt {d}})$$\\end{document} are principal, and ideal membership is characterised analytically. Calling an ideal in S′(Zd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}'(\\mathbb {Z}^\ exttt{d})$$\\end{document} fixed if there is a common index n∈Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{n}\\in \\mathbb {Z}^{\ exttt {d}}$$\\end{document} where each member vanishes, the fixed maximal ideals are described, and it is shown that not all maximal ideals are fixed. It is shown that finitely generated (hence principal) prime ideals in S′(Zd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}'(\\mathbb {Z}^{\ exttt {d}})$$\\end{document} are fixed maximal ideals. The Krull dimension of S′(Zd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}'(\\mathbb {Z}^{\ exttt {d}})$$\\end{document} is proved to be infinite, while the weak Krull dimension is shown to be equal to 1.

Flag Varieties of Type B/C and Twisted Yangian

We study the convolution algebra [Formula: see text] of [Formula: see text]-equivariant homology group on the Steinberg variety of type B/C and define an algebra [Formula: see text] that maps to [Formula: see text]. We also study the [Formula: see text]-equivariant case and prove that there is a map from a certain twisted current algebra to [Formula: see text].

A type C study of Braverman–Gaitsgory–Ginzburg’s construction of sln representations

In [Z. Fan, H. Ma and H. Xiao, Equivariant K-theory approach to [Formula: see text]-quantum groups, Publ. Res. Inst. Math. Sci. 58 (2022), no.3, 635–668], it is proved that the convolution algebra of the top Borel–Moore homology on the Steinberg variety of type [Formula: see text] realizes [Formula: see text], where [Formula: see text] is the fixed point subalgebra of involution on [Formula: see text]. So, the top Borel–Moore homology of the partial Springer fibers gives the representations of [Formula: see text]. In this paper, we study these representations using Schur–Weyl duality and Springer theory.

Weak$^*$-simplicity of convolution algebras on discrete groups

Weak$^*$-simplicity of convolution algebras on discrete groups

Lie groupoid deformations and convolution algebras

We define a morphism from the deformation complex of a Lie groupoid to the Hochschild complex of its convolution algebra, and show that it maps the class of a geometric deformation to the algebraic class of the induced deformation in Hochschild cohomology. Applied to the adiabatic groupoid, we show that the van Est map to deformation cohomology of Lie algebroids is induced by taking the classical limit of a quantization map on the dual of the Lie algebroid.

Field theories on ρ-deformed Minkowski space-time

We study one-loop perturbative properties of scalar field theories on the ρ-Minkowski space. The corresponding star-product, together with the involution are characterized from a combination of Weyl quantization and defining properties of the convolution algebra of the Euclidean group linked to the coordinate algebra of the ρ-Minkowski space. The natural integration measure linked to the Haar measure of the Euclidean group defines a trace for the star-product. One-loop properties of the 2-point and 4-point functions for families of complex-valued scalar field theories on ρ-Minkowski space are examined. For scalar theories with orientable interaction, the 2-point function is found to receive UV quadratically diverging one-loop corrections in 4 dimensions while no IR singularities generating UV/IR mixing appears. These however occur in the one-loop corrections to the 4-point function. As well, one-loop 2-point functions for theories with non-orientable interaction involve such IR singularities. These results are discussed.

Decay estimates for matrix coefficients of unitary representations of semisimple Lie groups

Let G be a connected semisimple Lie group with finite centre and K be a maximal compact subgroup thereof. Given a function u on G, we define Au to be the root-mean-square average over K, acting both on the left and the right, of u. We take a positive-real-valued spherical function ϕλ on G, and study the Banach convolution algebra of Cc(G)-functions u with the norm ‖u‖(λ):=∫GAu(x)ϕλ(x)dx. The C⁎ completion of this algebra is an exotic C⁎-algebra on G, in the sense that it lies “between” the reduced C⁎-algebra of G and the full C⁎-algebra of G, and in the sense that it arise as the completion of a star-algebra that does not contain an approximate identity.Using functional analysis and representation theory, we show that for all unitary representations π of G, there exists a unique minimal positive-real-valued spherical function ϕλ on G such that A〈π(⋅)ξ,η〉≤‖ξ‖Hπ‖η‖Hπϕλ. This estimate has nice features of both asymptotic pointwise estimates and Lebesgue space estimates; indeed it is equivalent to pointwise estimates |〈π(⋅)ξ,η〉|≤C(ξ,η)ϕλ for K-finite or smooth vectors ξ and η, and it exhibits different decay rates in different directions at infinity in G. Further, if we assume the latter inequality with arbitrary C(ξ,η), we can prove the former inequality and then return to the latter inequality with explicit knowledge of C(ξ,η). Finally, it holds everywhere in G, in contrast to asymptotic estimates which are not global and to Lp estimates which carry no pointwise information.

Catoids and modal convolution algebras

We show how modal quantales arise as convolution algebras Q^X of functions from catoids X, multisemigroups equipped with source and target maps, into modal quantales value or weight quantales Q. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in X, Q and Q^X. The catoids introduced generalise Schweizer and Sklar’s function systems and single-set categories to structures isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our correspondence results support a generic construction of weighted modal quantales from catoids. This construction is illustrated by many examples. We also relate our results to reasoning with stochastic matrices or probabilistic predicate transformers.

Exotic ideals in free transformation group C∗$\mathbf {C}^*$‐algebras

AbstractLet be a discrete group acting freely via homeomorphisms on the compact Hausdorff space and let be the completion of the convolution algebra with respect to a ‐norm . A non‐zero ideal is exotic if . We show that exotic ideals are present whenever is non‐amenable and there is an invariant probability measure on . This fact, along with the recent theory of exotic crossed product functors, allows us to provide answers to two questions of K. Thomsen.Using the Koopman representation and a recent theorem of Elek, we show that when is a countably‐infinite group having property (T) and is the Cantor set, there exists a free and minimal action of on and a ‐norm on such that contains the compact operators as an exotic ideal. We use this example to provide a positive answer to a question of A. Katavolos and V. Paulsen.The opaque and grey ideals in have trivial intersection with , and a result from Exel and Pitts [Characterizing groupoid ‐algebras of non‐Hausdorff étale groupoids, Lecture Notes in Mathematics, vol. 2306, Springer, Cham, 2022] shows that they coincide when the action of is free; however, the problem of whether these ideals can be non‐zero was left unresolved. We present an example of a free action of on a compact Hausdorff space along with a ‐norm for which these ideals are non‐trivial, in particular, they are exotic ideals.

On the Hochschild homology of proper Lie groupoids

We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and we prove that the Hochschild homology sheaf at each stalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolution algebra of its linearization, which is the transformation groupoid of a linear action of a compact isotropy group on a vector space. We then explain Brylinski’s ansatz to compute the Hochschild homology of the transformation groupoid of a compact group action on a manifold. We verify Brylinski’s conjecture for the case of smooth circle actions that the Hochschild homology is given by basic relative forms on the associated inertia space.

Approximately invertible elements in non-unital normed algebras

We introduce the concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian C*-algebras.

Strands algebras and the affine highest weight property for equivariant hypertoric categories

We show that the equivariant hypertoric convolution algebras introduced by Braden–Licata–Proudfoot–Webster are affine quasi hereditary in the sense of Kleshchev and compute the Ext groups between standard modules. Together with the main result of [27], this implies a number of new homological results about the bordered Floer algebras of Ozsváth–Szabó, including the existence of standard modules over these algebras. We prove that the Ext groups between standard modules are isomorphic to the homology of a variant of the Lipshitz–Ozsváth–Thurston bordered strands dg algebras.

Amenability of Monomial Algebras, Minimal Subshifts, and Free Subalgebras

Abstract We give a combinatorial characterization of amenability of monomial algebras and prove the existence of monomial Følner sequences, answering a question due to Ceccherini–Silberstein and Samet–Vaillant. We then use our characterization to prove that over projectively simple monomial algebras, every module is exhaustively amenable; we conclude that convolution algebras of minimal subshifts admit the same property. We deduce that any minimal subshift of positive entropy gives rise to a graded algebra, which does not satisfy an extension of Vershik’s conjecture on amenable groups, proposed by Bartholdi. Finally, we show that non-amenable monomial algebras must contain noncommutative free subalgebras. Examples are given to emphasize the sharpness and necessity of the assumptions in our results.

Singular subalgebroids

We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion of Lie subalgebroid by dropping the constant rank requirement. We lay the bases of a Lie theory for singular subalgebroids: we construct the associated holonomy groupoids, adapting the procedure of Androulidakis-Skandalis for singular foliations, in a way that keeps track of the choice of Lie groupoid integrating the ambient Lie algebroid. The holonomy groupoids are topological groupoids, and are suitable for noncommutative geometry as they allow for the construction of the associated convolution algebras. Further we carry out the construction for morphisms in a functorial way.

-invariant weighted Orlicz spaces

In this paper, we prove that σ-compactness of a locally compact group G is a necessary condition for the weighted Orlicz space to be a convolution algebra, and for a class of Orlicz spaces we give an equivalent condition. Also, we show that if is -invariant, then there exists a constant c > 0 such that Φ(ω(st)) ≤ c ω(s)ω Φ (t) for locally a.e. s, t ∈ G, where ω Φ is a locally integrable function satisfying an inclusion property.

Explicit Rieffel induction module for quantum groups

For \mathbb{G} an algebraic (or more generally, a bornological) quantum group and \mathbb{B} a closed quantum subgroup of \mathbb{G} , we build in this paper an induction module by explicitly defining, on the convolution algebra of \mathbb{G} , an inner product which takes its value in the convolution algebra of \mathbb{B} , as in the original approach of Rieffel. In this context, we study the link with the induction functor defined by Vaes. In the last part, we illustrate our result with parabolic induction of complex semisimple quantum groups. We first show that our induction functor coincides with the one already defined in the case of parabolic induction. Then we use the tools developed in this paper to give a geometric interpretation to the parabolic induction functor, following the approach suggested by Clare in the classical case.