Centralizing traces with automorphisms on triangular algebras

Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\), \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\)-center of \({\mathcal{T}}\). Suppose that \({\mathfrak{q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\)-bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}}\colon \mathcal{T}\longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\). The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}\in \mathcal{Z}_\xi(\mathcal{T})}\)) for all \({x\in \mathcal{T}}\). More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form.     

Semantical conditions for the definability of functions and relations

Let \({\mathcal{L}\subseteq \mathcal{L}^\prime}\) be first order languages, let \({R \in \mathcal{L}^\prime- \mathcal{L}}\) be a relation symbol, and let \({\mathcal{K}}\) be a class of \({\mathcal{L}^\prime}\)-structures. In this paper, we present semantical conditions equivalent to the existence of an \({\mathcal{L}}\)-formula \({\varphi(\vec{x})}\) such that \({\mathcal{K}\vDash \varphi(\vec{x}) \leftrightarrow R(\vec{x})}\), where \({\varphi}\) has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential Horn, etc.). For each of these definability results for relations, we also present an analogous version for the definability of functions. Several applications to natural definability questions in universal algebra have been included; most notably definability of principal congruences. The paper concludes with a look at term-interpolation in classes of structures with the same techniques used for definability. Here we obtain generalizations of two classical term-interpolation results: Pixley’s theorem for quasiprimal algebras, and the Baker–Pixley Theorem for finite algebras with a majority term.     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

Fourth-order Schrödinger type operator with singular potentials

We prove that a deformation of a hypersurface in an (n + 1)-dimensional real space form \({{\mathbb S}^{n+1}_{p,1}}\) induces a Hamiltonian variation of the normal congruence in the space \({{\mathbb L}({\mathbb S}^{n+1}_{p,1})}\) of oriented geodesics. As an application, we show that every Hamiltonian minimal submanifold in \({{\mathbb L}({\mathbb S}^{n+1})}\) (resp. \({{\mathbb L}({\mathbb H}^{n+1})}\)) with respect to the (para-)Kähler Einstein structure is locally the normal congruence of a hypersurface \({\Sigma}\) in \({{\mathbb S}^{n+1}}\) (resp. \({{\mathbb H}^{n+1}}\)) that is a critical point of the functional \({{\mathcal W}(\Sigma) = \int_\Sigma\left(\Pi_{i=1}^n|\epsilon+k_i^2|\right)^{1/2}}\), where k_i denote the principal curvatures of \({\Sigma}\) and \({\epsilon \in \{-1, 1\}}\). In addition, for \({n = 2}\), we prove that every Hamiltonian minimal surface in \({{\mathbb L}({\mathbb S}^{3})}\) (resp. \({{\mathbb L}({\mathbb H}^{3})}\)), with respect to the (para-)Kähler conformally flat structure, is the normal congruence of a surface in \({{\mathbb S}^{3}}\) (resp. \({{\mathbb H}^{3}}\)) that is a critical point of the functional \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K+1}}\) (resp. \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K-1}}\)), where H and K denote, respectively, the mean and Gaussian curvature of \({\Sigma}\).     

Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

The Proper Dissipative Extensions of a Dual Pair

Let A and \({(-\widetilde{A})}\) be dissipative operators on a Hilbert space \({\mathcal{H}}\) and let \({(A,\widetilde{A})}\) form a dual pair, i.e. \({A \subset \widetilde{A}^*}\), resp. \({\widetilde{A} \subset A^*}\). We present a method of determining the proper dissipative extensions \({\widehat{A}}\) of this dual pair, i.e. \({A\subset \widehat{A} \subset\widetilde{A}^*}\) provided that \({\mathcal{D}(A)\cap\mathcal{D}(\widetilde{A})}\) is dense in \({\mathcal{H}}\). Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed. Finally, we investigate the stability of the numerical range of the different dissipative extensions.     

Natural extensions and profinite completions of algebras

This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class \({\mathcal{A} = \mathbb{ISP}(\mathcal{M})}\), where \({\mathcal{M}}\) is a set, not necessarily finite, of finite algebras, it is shown that each \({{\bf A} \in \mathcal{A}}\) embeds as a topologically dense subalgebra of a topological algebra \({n_{\mathcal{A}}({\bf A})}\) (its natural extension), and that \({n_{\mathcal{A}}({\bf A})}\) is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that \({\mathcal{M}}\) is finite and \({\mathcal{A}}\) possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply.     

Joins and subdirect products of varieties

We generalise in three different directions two well-known results in universal algebra. Grätzer, Lakser and P?onka proved that independent subvarieties \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) of a variety \({\mathcal{V}}\) are disjoint and such that their join \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) (in the lattice of subvarieties of \({\mathcal{V}}\)) is their direct product \({\mathcal{V}_{1} \times \mathcal{V}_{2}}\) . Jónsson and Tsinakis provided a partial converse to this result: if \({\mathcal{V}}\) is congruence permutable and \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) are disjoint, then they are independent (and so \({\mathcal{V}_{1} \vee \mathcal{V}_{2} = \mathcal{V}_{1} \times \mathcal{V}_{2}}\)). We show that (i) if \({\mathcal{V}}\) is subtractive, then Jónsson’s and Tsinakis’ result holds under some minimal assumptions; (ii) if \({\mathcal{V}}\) satisfies some weakened permutability conditions, then disjointness implies a generalised notion of independence and \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) is the subdirect product of \({\mathcal{V}_{1}}\) and \({\mathcal{V}_2}\) ; (iii) the same holds if \({\mathcal{V}}\) is congruence 3-permutable.     

Gauge functions,Eikonal equations and Bôcher’s theorem on stratified Lie groups

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bôcher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).     

Multipliers of perturbed Cartesian products with an application to BSE-property

Given semisimple commutative Banach algebras \({\mathcal{A}}\) and \({\mathcal{B}}\) and a norm decreasing homomorphism \({\mathcal{T} : \mathcal{B} \rightarrow \mathcal{B}}\), we characterize the multipliers of the perturbed product Banach algebra \({\mathcal{A}\times_T \mathcal{B}}\). As an application it is shown that \({\mathcal{A}\times_T \mathcal{B}}\) has the Bochner–Schoenberg–Eberlein property if and only if both \({\mathcal{A}}\) and \({\mathcal{B}}\) have this property.     

Conformal embeddings of affine vertex algebras in minimal <Emphasis Type="Italic">W</Emphasis>-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine W-algebra \({W_k(\mathfrak{g}, \theta)}\) as a module for its maximal affine subalgebra \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) at a conformal level k, that is, whenever the Virasoro vectors of \({W_k(\mathfrak{g}, \theta)}\) and \({\mathscr{V}_k(\mathfrak{g}^\natural)}\) coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when \({\mathfrak{g}^{\natural}}\) is a semisimple Lie algebra, we show that, for a suitable conformal level k, \({W_k(\mathfrak{g}, \theta)}\) is isomorphic to an extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) by its simple module. We are able to prove that in certain cases \({W_k(\mathfrak{g}, \theta)}\) is a simple current extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\). In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra \({W_{k}(\mathit{sl}(4), \theta)}\) at k = ?8/3. We prove, as conjectured in [3], that \({W_{k}(\mathit{sl}(4), \theta)}\) is isomorphic to the vertex algebra \({\mathscr{R}^{(3)}}\), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra \({V_k (\mathit{sl}(n))}\) at certain admissible levels and for \({V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}\) at arbitrary levels.     

Spectral properties and stability of perturbed Cartesian product

Let \({\mathcal {A}}\) and \({\mathcal {B}}\) be commutative Banach algebras, and let \(T:{\mathcal {B}} \rightarrow {\mathcal {A}}\) be an algebra homomorphism with \({\Vert T\Vert }\le 1\). Then T induces a Banach algebra product \(\times _T\) perturbing the coordinatewise product on the Cartesian product space \({\mathcal {A}} \times {\mathcal {B}}\). We show that the spectral properties like spectral extension property, unique uniform norm property, regularity, weak regularity as well as Ditkin’s condition are stable with respect to this product.     

Mutually prime sequences of inner functions and the ranks of subspaces over the bidisk

Let \({\{\varphi_n(z)\}_{n\ge0}}\) be a sequence of inner functions satisfying that \({\zeta_n(z):=\varphi_n(z)/\varphi_{n+1}(z)\in H^\infty(z)}\) for every n ≥ 0 and \({\{\varphi_n(z)\}_{n\ge0}}\) have no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace \({\mathcal{M}}\) of \({H^2(\mathbb{D}^2)}\) . We write \({\mathcal{N}= H^2(\mathbb{D}^2)\ominus\mathcal{M}}\) . If \({\{\zeta_n(z)\}_{n\ge0}}\) ia a mutually prime sequence, then we shall prove that \({rank_{\{T^\ast_z,T^\ast_w\}} \mathcal{N}=1}\) and \({rank_{\{\mathcal{F}^\ast_z\}}(\mathcal{M}\ominus w\mathcal{M})=1}\) , where \({\mathcal{F}_z}\) is the fringe operator on \({\mathcal{M}\ominus w\mathcal{M}}\) .     

Spherically Balanced Hilbert Spaces of Formal Power Series in Several Variables-II

We continue the study of spherically balanced Hilbert spaces initiated in the first part of this paper. Recall that the complex Hilbert space \(H^2(\beta )\) of formal power series in the variables \(z_1, \ldots , z_m\) is spherically balanced if and only if there exist a Reinhardt measure \(\mu \) supported on the unit sphere \(\partial {\mathbb {B}}\) and a Hilbert space \(H^2(\gamma )\) of formal power series in the variable \(t\) such that

$$\begin{aligned} \Vert f\Vert ^2_{H^2(\beta )} = \int _{\partial {\mathbb {B}}}\Vert {f_z}\Vert ^2_{H^2(\gamma )}~d\mu (z)~(f \in H^2(\beta )), \end{aligned}$$

where \(f_z(t)=f(t z)\) is a formal power series in the variable \(t\). In the first half of this paper, we discuss operator theory in spherically balanced Hilbert spaces. The first main result in this part describes quasi-similarity orbit of multiplication tuple \(M_z\) on a spherically balanced space \(H^2(\beta ).\) We also observe that all spherical contractive multi-shifts on spherically balanced spaces admit the classical von Neumann’s inequality. In the second half, we introduce and study a class of Hilbert spaces, to be referred to as \({\mathcal {G}}\)-balanced Hilbert spaces, where \({\mathcal {G}}={\mathcal {U}}(r_1) \times {\mathcal {U}}(r_2) \times \cdots \times {\mathcal {U}}(r_k)\) is a subgroup of \({\mathcal {U}}(m)\) with \(r_1 + \cdots + r_k=m.\) In the case in which \({\mathcal {G}}={\mathcal {U}}(m),\) \({\mathcal {G}}\)-balanced spaces are precisely spherically balanced Hilbert spaces.

     

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

We consider various aspects of the Segre variety \({\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}\) in PG(7, 2), whose stabilizer group \({\mathcal{G}_{\mathcal{S}}<{\rm GL}(8,2)}\) has the structure \({\mathcal{N}\rtimes{\rm Sym}(3),}\) where \({\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}\) In particular we prove that \({\mathcal{S}}\) determines a distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2)}\) such that \({A\mathcal{Z}A^{-1}=\mathcal{Z},}\) for all \({A\in\mathcal{G}_{\mathcal{S}},}\) and in consequence \({\mathcal{S}}\) determines a \({\mathcal{G}_{\mathcal{S}}}\)-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties \({\mathcal{S}_{1,1,1}(2)}\) in PG(7, 2) come along in triplets \({\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}\) which share the same distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2).}\) We conclude by determining all fifteen \({\mathcal{G}_{\mathcal{S}}}\)-invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five \({\mathcal{G}_{\mathcal{S}}}\)-orbits of points in PG(7, 2).     

Proximally well-monotone covers and QH-singularity

We use a certain class of well-monotone covers on a quasi-uniform space \({(X, \mathcal{U})}\) to investigate whether there are quasi-uniformities \({\mathcal{V}}\) that are distinct from \({\mathcal{U}}\), but have the property that the associated Hausdorff quasi-uniformities \({\mathcal{U}_H}\) and \({\mathcal{V}_H}\) on the hyperspace of X have the same underlying topologies.     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

Compact right multipliers on a Banach algebra related to locally compact semigroups

For a locally compact semigroup \({\mathcal{S}}\), let \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))\) be the Banach space of all μ-measurable (\(\mu\in M_{a}({\mathcal{S}})\)) functions vanishing at infinity, where \(M_{a}({\mathcal{S}})\) denotes the algebra of all measures in the measure algebra \(M({\mathcal{S}})\) of \({\mathcal{S}}\) with continuous translations. Here, we study right compact multipliers on the Banach algebra \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))^{*}\) equipped with an Arens product.