Convolution Algebra Research Articles

Convolution algebra of superoperators and nonseparability witnesses for quantum operations

We define a product between quantum superoperators which is preserved under the Choi–Jamiołkowski–Kraus–Sudarshan channel-state isomorphism. We then identify the product as the convolution on the space of superoperators, with respect to which the channel-state duality is also an algebra isomorphism. We find that any witness operator for detecting nonseparability of quantum operations on separated parties can be written entirely within the space of superoperators with the help of the convolution product.

Symmetry and spectral invariance for topologically graded C∗$C^*$‐algebras and partial action systems

A discrete group G ${\sf G}$ is called rigidly symmetric if the projective tensor product between the convolution algebra ℓ 1 ( G ) $\ell ^1({\sf G})$ and any C ∗ $C^*$ -algebra A $\mathcal {A}$ is symmetric. We show that in each topologically graded C ∗ $C^*$ -algebra over a rigidly symmetric group there is a ℓ 1 $\ell ^1$ -type symmetric Banach ∗ $^*$ -algebra, which is inverse closed in the C ∗ $C^*$ -algebra. This includes new general classes, as algebras admitting dual actions and partial crossed products. Results including convolution dominated kernels, inverse closedness with respect with ideals or weighted versions of the ℓ 1 $\ell ^1$ -decay are included. Various concrete examples are presented.

Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types

For a Dynkin quiver Q Q (of type A D E \mathrm {ADE} ), we consider a central completion of the convolution algebra of the equivariant K K -group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc’s monoidal category C Q \mathcal {C}_{Q} of modules over the quantum loop algebra U q ( L g ) U_{q}(L\mathfrak {g}) via Nakajima’s homomorphism. As an application, we show that Kang-Kashiwara-Kim’s generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with Q Q and Hernandez-Leclerc’s category C Q \mathcal {C}_{Q} , assuming the simpleness of some poles of normalized R R -matrices for type E \mathrm {E} .

Non-spherical Harish-Chandra Fourier Transforms on Real Reductive Groups

The Harish-Chandra Fourier transform, $$f\mapsto \mathcal {H}f,$$ is a linear topological algebra isomorphism of the spherical (Schwartz) convolution algebra $$\mathcal {C}^{p}(G//K)$$ (where K is a maximal compact subgroup of any arbitrarily chosen group G in the Harish-Chandra class and $$0<p\le 2$$ ) onto the (Schwartz) multiplication algebra $$\bar{\mathcal {Z}}({\mathfrak {F}}^{\epsilon })$$ (of $$\mathfrak {w}$$ -invariant members of $$\mathcal {Z}({\mathfrak {F}}^{\epsilon }),$$ with $$\epsilon =(2/p)-1$$ ). This is the well-known Trombi–Varadarajan theorem for spherical functions on the real reductive group, G. Even though $$\mathcal {C}^{p}(G//K)$$ is a closed subalgebra of $$\mathcal {C}^{p}(G),$$ a similar theorem has not however been successfully proved for the full Schwartz convolution algebra $$\mathcal {C}^{p}(G)$$ except; for $$\mathcal {C}^{p}(G/K)$$ (whose method is essentially that of Trombi–Varadarajan, as shown by M. Eguchi); for few specific examples of groups (notably $$G=SL(2,\mathbb {R})$$ ) and; for some notable values of p (with restrictions on G and/or on members of $$\;\mathcal {C}^{p}(G)$$ ). In this paper, we construct an appropriate image of the Harish-Chandra Fourier transform for the full Schwartz convolution algebra $$\mathcal {C}^{p}(G),$$ without any restriction on any of G, p and members of $$\;\mathcal {C}^{p}(G).$$ Our proof, that the Harish-Chandra Fourier transform, $$f\mapsto \mathcal {H}f,$$ is a linear topological algebra isomorphism on $$\mathcal {C}^{p}(G),$$ equally shows that its image $$\mathcal {C}^{p}(\widehat{G})$$ can be nicely decomposed, that the full invariant harmonic analysis is available and implies that the definition of the Harish-Chandra Fourier transform may now be extended to include all p-tempered distributions on G and to the zero-Schwartz spaces.

Isomorphisms of algebras of convolution operators

For $p,q\in [1,\infty)$, we study the isomorphism problem for the $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its group von Neumann algebra, we show that this is the case for the algebras $\mathrm{CV}_p(G)$ of $p$-convolvers and $\mathrm{PM}_p(G)$ of $p$-pseudomeasures, for $p\neq 2$. More generally, we show that if $\mathrm{CV}_p(G)$ is isometrically isomorphic to $\mathrm{CV}_q(H)$, with $p,q\neq 2$, then $G$ must be isomorphic to $H$ and $p$ and $q$ are either equal or conjugate. This implies that there is no $L^p$-version of Connes' uniqueness of the hyperfinite II$_1$-factor. Similar results apply to the algebra $\mathrm{PF}_p(G)$ of $p$-pseudofunctions, generalizing a classical result of Wendel. We also show that other $L^p$-rigidity results for groups can be easily recovered and extended using our main theorem. Our results answer questions originally formulated in the work of Herz in the 70's. Moreover, our methods reveal new information about the Banach algebras in question. As a non-trivial application, we verify the reflexivity conjecture for all Banach algebras lying between $\mathrm{PF}_p(G)$ and $\mathrm{CV}_p(G)$: if any such algebra is reflexive and amenable, then $G$ is finite.

Motivic Springer theory

We show that representations of convolution algebras such as Lustzig’s graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in type A and A˜ can be realized in terms of certain equivariant motivic sheaves called Springer motives. To this end, we lay foundations to a motivic Springer theory and prove formality results using weight structures.As byproduct, we express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in type A˜ (with cyclic orientation) admit an affine paving.

A PARAMETRIZATION OF δ-SPHERICAL FUNCTIONS ON COMMUTATIVE TRIPLES ASSOCIATED WITH NILPOTENT LIE GROUPS

Let $N$ be a connected and simply connected nilpotent Lie group, $K$ be a compact subgroup of $Aut(N)$, the group of automorphisms of $N$ and $\delta$ be a class of unitary irreducible representations of $K$. The triple $(N,K,\delta)$ is a commutative triple if the convolution algebra $\mathfrak{U}_{\delta}^{1}(N)$ of $\delta$-radial integrable functions is commutative. In this paper, we obtain first a parametrization of $\delta$ spherical functions by means of the unitary dual $\widehat{N}$ and then an inversion formula for the spherical transform of $F\in \mathfrak{U}_{\delta}^{1}(N)$.

Convolution algebras for relational groupoids and reduction

We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group $G$ and a given normal subgroup $H$. We also give conditions for the existence of a Haar system of measures on a relational groupoid compatible with the convolution, and we prove a reduction theorem that recovers the usual convolution of a Lie groupoid.

Convolution and concurrency

AbstractWe show how concurrent quantales and concurrent Kleene algebras arise as convolution algebras of functions from relational structures with two ternary relations that satisfy relational interchange laws into concurrent quantales or Kleene algebras, among others. The elements of the quantales can be understood as weights; the case where weights are drawn from the booleans corresponds to languages. We develop a correspondence theory between properties of the relational structures and algebraic properties in the weight and convolution algebras in the sense of modal and substructural logics, or boolean algebras with operators. The resulting correspondence triangles yield in particular general construction principles for models of concurrent quantales and Kleene algebras as convolution algebras from much simpler relational structures, including weighted ones for quantitative applications. As examples, we construct the concurrent quantales and Kleene algebras of weighted words, digraphs, posets, isomorphism classes of finite digraphs and pomsets.

Fractional glassy relaxation and convolution modules of distributions

Solving fractional relaxation equations requires precisely characterized domains of definition for applications of fractional differential and integral operators. Determining these domains has been a longstanding problem. Applications in physics and engineering typically require extension from domains of functions to domains of distributions. In this work convolution modules are constructed for given sets of distributions that generate distributional convolution algebras. Convolutional inversion of fractional equations leads to a broad class of multinomial Mittag-Leffler type distributions. A comprehensive asymptotic analysis of these is carried out. Combined with the module construction the asymptotic analysis yields domains of distributions, that guarantee existence and uniqueness of solutions to fractional differential equations. The mathematical results are applied to anomalous dielectric relaxation in glasses. An analytic expression for the frequency dependent dielectric susceptibility is applied to broadband spectra of glycerol. This application reveals a temperature independent and universal dynamical scaling exponent.

Spectral Invariance of *-Representations of Twisted Convolution Algebras with Applications in Gabor Analysis

We show spectral invariance for faithful *-representations for a class of twisted convolution algebras. More precisely, if G is a locally compact group with a continuous 2-cocycle c for which the corresponding Mackey group G_c is C^*-unique and symmetric, then the twisted convolution algebra L^1 (G,c) is spectrally invariant in {mathbb {B}}({mathcal {H}}) for any faithful *-representation of L^1 (G,c) as bounded operators on a Hilbert space {mathcal {H}}. As an application of this result we give a proof of the statement that if Delta is a closed cocompact subgroup of the phase space of a locally compact abelian group G', and if g is some function in the Feichtinger algebra S_0 (G') that generates a Gabor frame for L^2 (G') over Delta , then both the canonical dual atom and the canonical tight atom associated to g are also in S_0 (G'). We do this without the use of periodization techniques from Gabor analysis.

A unified construction of product formulas and convolutions for Sturm–Liouville operators

We establish a positive product formula for the solutions of the Sturm–Liouville equation $$\ell (u) = \lambda u$$ , where $$\ell $$ belongs to a general class which includes singular and degenerate Sturm–Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm–Liouville operator. We show that the product formula gives rise to a probability preserving convolution algebra structure on the space of finite measures which satisfies the basic axioms for developing harmonic analysis on the convolution algebra. Unlike previous works, our framework includes a subfamily of Sturm–Liouville operators for which the support of the convolution of Dirac measures is noncompact. The connection with hypergroup theory is discussed. Convolution-type integral equations on weighted Lebesgue spaces are also discussed, and a solvability condition is established.

The convolution algebra of Schwartz kernels along a singular foliation

Motivated by the study of H\"ormander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized functions on the underlying manifold, and on the leaves and their holonomy covers. This generalizes Schwartz kernel operators to singular foliations. We also define the algebra of smoothing operators in this context and prove that it is a two-sided ideal.

On actions of connected Hopf algebras

Let H H be a connected Hopf algebra acting on an algebra A A . Working over a base field having characteristic 0 0 , we show that for a given prime (semi-prime, completely prime) ideal I I of A A , the largest H H -stable ideal of A contained in I I is also prime (semi-prime, completely prime). We also prove a similar result for certain subrings of convolution algebras.

Quantum Stochastic Products and the Quantum Convolution

A quantum stochastic product is a binary operation on the space of quantum states preserving the convex structure. We describe a class of associative stochastic products, the twirled products, that have interesting connections with quantum measurement theory. Constructing such a product involves a square integrable group representation, a probability measure and a fiducial state. By extending a twirled product to the full space of trace class operators, one obtains a Banach algebra. This algebra is commutative if the underlying group is abelian. In the case of the group of translations on phase space, one gets a quantum convolution algebra, a quantum counterpart of the classical phase-space convolution algebra. The peculiar role of the fiducial state characterizing each quantum convolution product is highlighted.

The Mirković–Vilonen basis and Duistermaat–Heckman measures

Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are perfect, i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson-Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated to an MV basis element equals the Duistermaat-Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig's dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6.

Real Springer fibers and odd arc algebras

We give a topological description of the two-row Springer fiber over the real numbers. We show its cohomology ring coincides with the oddification of the cohomology ring of the complex Springer fiber introduced by Lauda-Russell. We also realize Ozsv\'ath-Rasmussen-Szab\'o odd TQFT from pullbacks and exceptional pushforwards along inclusion and projection maps between hypertori. Using these results, we construct the odd arc algebra as a convolution algebra over components of the real Springer fiber, giving an odd analogue of a construction of Stroppel-Webster.

Convolution algebra for extended Feller convolution

We apply the recently introduced framework of admissible homomorphisms in the form of a convolution algebra of $$\mathbb{C}^2$$ -valued admissible homomorphisms to handle two-dimensional uni-directional homogeneous stochastic kernels. The algebra product is a non-commutative extension of the Feller convolution needed for an adequate operator representation of such kernels: a pair of homogeneous transition functions uni-directionally intertwined by the extended Chapman–Kolmogorov equation is a convolution empathy; the associated Fokker–Planck equations are re-written as an implicit Cauchy equation expressed in terms of admissible homomorphisms. The conditions of solvability of such implicit evolution equations follow from the consideration of generators of a convolution empathy.

Factorization in Denjoy-Carleman classes associated to representations of [formula omitted

For two types of moderate growth representations of (Rd,+) on sequentially complete locally convex Hausdorff spaces (including F-representations [14]), we introduce Denjoy-Carleman classes of ultradifferentiable vectors and show a strong factorization theorem of Dixmier-Malliavin type for them. In particular, our factorization theorem solves [14, Conjecture 6.4] for analytic vectors of representations of G=(Rd,+). As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property.

Yang–Baxter algebras, convolution algebras, and Grassmannians

This paper surveys a new actively developing direction in contemporary mathematics which connects quantum integrable models with the Schubert calculus for quiver varieties: there is a purely geometric construction of solutions to the Yang–Baxter equation and their associated Yang–Baxter algebras which play a central role in quantum integrable systems and exactly solvable (integrable) lattice models in statistical physics. A simple but explicit example is given using the classical geometry of Grassmannians in order to explain some of the main ideas. The degenerate five-vertex limit of the asymmetric six-vertex model is considered, and its associated Yang–Baxter algebra is identified with a convolution algebra arising from the equivariant Schubert calculus of Grassmannians. It is also shown how our methods can be used to construct quotients of the universal enveloping algebra of the current algebra (so-called Schur-type algebras) acting on the tensor product of copies of its evaluation representation . Finally, our construction is connected with the cohomological Hall algebra for the -quiver. Bibliography: 125 titles.