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Introverted algebras with mean value

Let A be an introverted algebra with mean value. We prove that its spectrum \Delta (A) is a compact topological semigroup, and that the kernel K(\Delta (A)) of \Delta (A) is a compact topological group over which the mean value on A can be identified as the Haar integral. Based on these facts and also on the fact that K(\Delta (A)) is an ideal of \Delta (A), we define the convolution over \Delta (A). We then use it to derive some new convergence results involving the convolution product of sequences. These convergence results provide us with an efficient method for studying the asymptotics of nonlocal problems. The obtained results systematically establish the connection between the abstract harmonic analysis and the homogenization theory. To illustrate this, we work out some homogenization problems in connection with nonlocal partial differential equations.


Introduction and the main results
Let A be an algebra with mean value on R N , that is, a closed subalgebra of the commutative Banach algebra BUC(R N ) (of bounded uniformly continuous functions on R N ) that contains the constants, is closed under complex conjugation, is translation invariant and has an invariant mean value. Thus A is a commutative Banach algebra with spectrum denoted by ∆(A). We consider each element of ∆(A) as a multiplicative linear functional on A. The usual (or Gelfand) topology of ∆(A) is the relative weak * topology induced on ∆(A) by σ(A ′ , A). The properties of the Gelfand space ∆(A) are well known and can be found in any text book about Banach algebras, see for instance [15]. The commonly known property is that ∆(A) is a compact topological space. It is also known that it becomes metrizable provided that A is separable. In some special cases, ∆(A) is well characterized. For example, when A is the algebra of almost periodic functions, ∆(A) is a compact topological abelian group, and in particular if A is the algebra of periodic functions, ∆(A) is the N -dimensional torus T N .
On the other hand, it seems that almost nothing is known about ∆(A) for general algebras with mean value A. In this paper our aim is to characterize the space ∆(A) for some general algebras A and present some applications. The relevance and importance of this characterization is due, among other things, to the fact that if A is introverted [19, p. 121] (see also [7, p. 540] for the general concept of introversion) then ∆(A) is a compact topological semigroup (see Theorem 2 below). We are particularly interested in the following questions: In this paper we try to answer these questions and present some applications of the obtained results, and the notions introduced to study them. In particular we test these questions on the algebra of almost periodic functions, the algebra of functions that converge at infinity, and in general on any closed subalgebra of the algebra of weakly almost periodic functions.
The content of the paper is summarized as follows. Section 2 deals with the fundamental notions about algebras with mean value, the generalized Besicovitch spaces and the derivation theory both associated to them. In Section 3 we study the introverted algebras with mean value. We give the answer to the questions (a), (b) and (c) raised above, but this in the special setting of introverted algebras with mean value. The main results of this section are as follows.
(1) If the algebra A is introverted, then ∆(A) is a compact topological semigroup. This result is known in the general theory of Banach algebras of uniformly continuous functions; see e.g. [16,19]. Our proof relies only on the compactness of ∆(A). We also show that, if further the multiplication defined on ∆(A) is jointly continuous, then ∆(A) is a compact topological group. The second main result of Section 3 reads as (2) If A is introverted, then A is a subalgebra of the weakly almost periodic functions, and moreover, if the multiplication in ∆(A) is jointly continuous, then A is a subalgebra of the almost periodic functions. The third main result of Section 3 is the answer to the question (c) raised above, and is this (3) If A is introverted, then the kernel K(∆(A)) of ∆(A) is a compact topological group, and the mean value on A can be identified as the Haar integral over K(∆(A)).
In all the previous works dealing with algebras with mean value, the mean value were identified as the integral over the spectrum only, see for instance [21,22,27]. Here we go further and we will see that this result is of first importance when defining the convolution over the spectrum of such algebras. The last main result of Section 3 is the basic tool that enables us to establish the connection between the convolution and the Σ-convergence method. It is new and constitutes the point of departure of all the results dealing with convergence of sequences involving delay. It reads as (4) Let A be an introverted algebra with mean value on R N . Then if δ y denotes the Dirac mass at y, we have δ y ∈ K(∆(A)) for almost all y ∈ R N . We end Section 3 with the answer to question (d). We define the convolution on the spectrum ∆(A) in terms of its kernel K(∆(A)).
In Section 4, in order to deal with some applications in homogenization theory, we state and prove a De Rham type result. More precisely, the main result of this section is this (5) If A is an algebra with mean value on R N and L is a bounded linear functional on (B 1,p ′ A ) N which vanishes on the kernel of the divergence, then there exists a function f ∈ B p A such that L = ∇ y f . Although classically known in the general framework of the distribution theory and in the special setting of periodic functions, the above result is new in the framework of general algebras with mean value.
In Section 5 we gather the notation and basic facts we need about the Σconvergence method. Section 6 deals with the answer to question (e). We study therein the connection between the convolution and the Σ-convergence method. The main result of this section is the following.
One of the main motivation of the present study arises from the importance and applications of the phenomena with delay in the real life. The results of Section 6 have applications in mathematical neuroscience, engineering sciences etc. In order to show how it works, we present in Sections 7 and 8 two applications of the results of the previous sections to homogenization theory. In particular in Section 8, we give the answer to a question raised by Attouch and Damlamian [2] about the homogenization of nonlinear operators involving convolution. This is a true advance as far as the comprehension of the spectrum of an algebra with mean value as well as the homogenization theory in general, are concerned.
We end this section with some preliminary notions. A directed set is a set E equipped with a binary relation such that • α α for all α ∈ E; • if α β and β γ then α γ; • for any α, β ∈ E there exists γ ∈ E such that α γ and β γ.
A net in a set X is a mapping α → x α from a directed set E into X. We usually denote such a mapping by (x α ) α∈E , or just by (x α ) if E is understood, and we say that (x α ) is indexed by E. Here below are some examples of directed sets: (i) The set of positive integers N, with j k if and only if j ≤ k.
(ii) The set R\{a} (a ∈ R), with x y if and only if |x − a| ≥ |y − a|. Unless otherwise stated, vector spaces throughout are assumed to be real vector spaces, and scalar functions are assumed to take real values. The results obtained here easily carry over mutatis mutandis to the complex setting.
The results of Section 6 were announced in [31].

Fundamentals of algebras with mean value
We refer the reader to [38] for details regarding some of the results of this section. A closed subalgebra A of the C*-algebra of bounded uniformly continuous functions BUC(R N ) is an algebra with mean value on R N [14,27,41] if it contains the constants, is translation invariant (u(· + a) ∈ A for any u ∈ A and each a ∈ R N ) and is such that any of its elements possesses a mean value, that is, for any u ∈ A, the sequence (u ε ) ε>0 (defined by u ε (x) = u(x/ε), x ∈ R N ) weakly * -converges in L ∞ (R N ) to some constant real function M (u) as ε → 0.
It is known that A (endowed with the sup norm topology) is a commutative C*-algebra with identity. We denote by ∆(A) the spectrum of A and by G the Gelfand transformation on A. We recall that ∆(A) (a subset of the topological dual A ′ of A) is the set of all nonzero multiplicative linear functionals on A, and G is the mapping of A into C(∆(A)) such that G(u)(s) = s, u (s ∈ ∆(A)), where , denotes the duality pairing between A ′ and A. When equipped with the relative weak * topology on A ′ (the topological dual of A), ∆(A) is a compact topological space, and the Gelfand transformation G is an isometric * -isomorphism identifying A with C(∆(A)) as C*-algebras. Moreover the mean value M defined on A is a nonnegative continuous linear functional that can be expressed in terms of a Radon measure β (of total mass 1) in ∆(A) (called the M -measure for A [21]) satisfying the property that M (u) = ∆(A) G(u)dβ for u ∈ A.
To any algebra with mean value A we associate the following subspaces: From this last property one may naturally define the space B ∞ A as follows: We endow B ∞ A with the seminorm [f ] ∞ = sup 1≤p<∞ f p , which makes it a complete seminormed space. We recall that the spaces B p A (1 ≤ p ≤ ∞) are not in general Fréchet spaces since they are not separated in general. The following properties are worth noticing [22,27]: (1) The Gelfand transformation G : A → C(∆(A)) extends by continuity to a unique continuous linear mapping (still denoted by G) of B p A into L p (∆(A)), which in turn induces an isometric isomorphism (2) The mean value M defined on A, extends by continuity to a positive continuous linear form (still denoted by and all a ∈ R N , where τ a u = u(· + a). Moreover for u ∈ B p A we have u p = [M (|u| p )] 1/p , and for u + N ∈ B p A we may still define its mean value once again denoted by M , as M (u + N ) = M (u).
Let 1 ≤ p ≤ ∞. In order to define the Sobolev type spaces associated to the algebra A, we consider the N -parameter group of isometries {T (y) : y ∈ R N } defined by Since the elements of A are uniformly continuous, {T (y) : y ∈ R N } is a strongly continuous group of operators in L(B p A , B p A ) (the Banach space of continuous linear We also associate to {T (y) : y ∈ R N } the following N -parameter group {T (y) : y ∈ R N } defined by A . The group {T (y) : y ∈ R N } is also strongly continuous. The infinitesimal generator of T (y) (resp. T (y)) along the ith coordinate direction, denoted by D i,p (resp. ∂ i,p ), is defined as where here we have used the same letter u to denote the equivalence class of an The following results are justified in [38]. We refer the reader to the above-mentioned paper for their justification.
). The next result allows us to see D i,p as a generalization of the usual partial derivative.
From (2.1) we infer that D i,p • ̺ = ̺ • ∂/∂y i , that is, D i,p generalizes the usual partial derivative ∂/∂y i . One may also define higher order derivatives by setting The counter-part of the above properties also holds with The following relation between D i,p and ∂ i,p holds.
. Then the inequality for a positive constant c independent of u and t, yields and we have the Proposition 2 ([38, Proposition 4]). The following assertions hold.
The formula (iii) in the above proposition leads to the equality This suggests us to define the concepts of distributions on A and of a weak derivative. Before we can do that, let us endow D A (R N ) = ̺(A ∞ ) with its natural topology defined by the family of norms N n (u) = sup |α|≤n sup y∈R N |D α ∞ u(y)|, n ∈ N. In this topology, D A (R N ) is a Fréchet space. We denote by D ′ A (R N ) the topological dual of D A (R N ). We endow it with the strong dual topology. The elements of D ′ A (R N ) are called the distributions on A. One can also define the weak derivative of f ∈ D ′ A (R N ) as follows: for any α ∈ N N , D α f stands for the distribution defined by the formula with continuous embedding, so that one may define the weak derivative of any f ∈ B p A , and it verifies the following functional equation: In particular, for f ∈ D i,p we have We are therefore justified in saying that B 1,p A is a Banach space under the norm · B 1,p A . The same result holds for W 1,p (∆(A)).
Moreover it is a fact that D A (R N ) (resp. D(∆(A))) is a dense subspace of B 1,p A (resp. W 1,p (∆(A))).

We need a further notion. A function f ∈ B 1
A is said to be invariant if for any y ∈ R N , T (y)f = f . It is immediate that the above notion of invariance is the well-known one relative to dynamical systems. An algebra with mean value will therefore said to be ergodic if every invariant function f is constant in B 1 A . As in [6] one may show that f ∈ B 1 A is invariant if and only if D i,1 f = 0 for all 1 ≤ i ≤ N . We denote by I p A the set of f ∈ B p A that are invariant. The set I p A is a closed vector subspace of B p A satisfying the following important property: We therefore endow B 1,p A /I p A with the seminorm which, in view of (2.2) is actually a norm under which . We define the gradient operator D p u and the divergence operator div p v by We define also the Laplacian operator by ∆ p u = div p (D p u), and it holds that ∆ p ̺(u) = ̺(∆ y u) for all u ∈ A ∞ , where ∆ y denotes the usual Laplacian operator on R N y . The following obvious properties are satisfied. 1. The divergence operator div p ′ (p ′ = p/(p − 1)) sends continuously and The same definitions and properties hold true when replacing B p A by L p (∆(A)), D p by ∂ p , div p by div p and ∆ p by ∆ p , and we have the following relations between these operators (provided that they make sense): Before we can state one of the most important result of this section, we still need to make some preliminaries and some notation. To this end let f ∈ B p A . We know that D αi f exists (in the sense of distributions) and that D αi f = D i,p f if f ∈ D i,p . So we can drop the subscript p and therefore denote D i,p (resp. ∂ i,p ) by ∂/∂y i (resp. ∂ i ). Thus, D y ≡ ∇ y will stand for the gradient operator (∂/∂y i ) 1≤i≤N and div y for the divergence operator div p , with G 1 • div y = div. We will also denote ∂ ≡ (∂ 1 , ..., ∂ N ). Finally, we shall denote the Laplacian operator on B p A by ∆ y .

Introverted algebras with mean value
For useful purposes, we need to characterize the spectrum of an algebra with mean value A. It is known to be the Stone-Čech compactification of R N provided that A separates the points of R N as seen below. Proof. For each y ∈ R N let us define δ y , the Dirac mass at y by setting δ y (u) = u(y) for u ∈ A. Then the mapping δ : y → δ y , of R N into ∆(A), is continuous and has dense range. In fact as the topology in ∆(A) is the weak * one and further the mappings y → δ y (u) = u(y), u ∈ A, are continuous on R N , it follows that δ is continuous. Now assuming that δ(R N ) is not dense in ∆(A) we derive the existence of a non empty open subset U of ∆(A) such that U ∩ δ(R N ) = ∅. Then by Urysohn's lemma there exists v ∈ C(∆(A)) with v = 0 and v| ∆(A))\U = 0 where v| ∆(A))\U denotes the restriction of v to ∆(A)\U . By the Gelfand representation theorem, v = G(u) for some u ∈ A. But then Next, every f in A (viewed as element of B(R N )) extends continuously to ∆(A) in the sense that there exists f ∈ C(∆(A)) such that f (δ y ) = f (y) for all y ∈ R N (just take f = G(f )). Finally assume that A separates the points of R N . Then the mapping δ : R N → δ(R N ) is a homeomorphism. In fact, we only need to prove that δ is injective. For that, let y, z ∈ R N with y = z; since A separates the points of R N , there exists a function u ∈ A such that u(y) = u(z), hence δ y = δ z , and our claim is justified. We therefore conclude that the couple (∆(A), δ) is the Stone-Čech compactification of R N .
Before dealing with the general situation, let us begin with a summary of the facts and notation we shall use in connection with the semigroup theory.
A semigroup is a set supplied with an associative binary operation that will be referred to as multiplication. Let S be a semigroup with multiplication S × S → S, (x, y) → xy. If S is a topological space, the multiplication is said to be separately continuous if the maps x → xy and x → yx (for each fixed y in S) of S into itself, are continuous. The multiplication in S is said to be jointly continuous if the map (x, y) → xy is continuous.
Let S be a semigroup with and identity e (ex = xe = x, all x ∈ S). S is said to be a topological semigroup if S is a Hausdorff topological space in which the multiplication is separately continuous. There is a vast literature on topological semigroups with the stronger assumption that the multiplication is jointly continuous; see e.g. [11,34]. Although our results do work in that special setting, it is not necessary for us to consider such kind of semigroups for some obvious reasons that shall be given later (see e.g. Remark 2). Now, let A be an algebra with mean value on R N . For µ ∈ ∆(A) and f ∈ A we define the function Then since A is translation invariant, T µ f is well defined as an element of BUC(R N ). This defines a bounded linear operator T µ : . This concept is due to T. Mitchell [19, p. 121] (in which he rather uses the symbol M -introverted instead of merely introverted) and has applications to fixed points theorems [19]. Here below is the characterization of the concept of introversion of an algebra with mean value A.

Lemma 3. Let A be an algebra with mean value on R N . Then A is introverted if and only if for any
, hence τ aν f → g pointwise, i.e. g belongs to the pointwise closure of the orbit of f . Whence g ∈ A, and so A is introverted.
As a consequence of the preceding lemma, some examples of introverted algebras are quoted below. Proof. That these algebras are algebras with mean value is well known; see e.g. [22,27]. B ∞ (R N ) and AP(R N ) are both closed subalgebras of WAP(R N ) (see [10]). Thus, to see that B ∞ (R N ), AP(R N ) and WAP(R N ) are introverted, it is sufficient to show that any closed translation invariant subalgebra A of WAP(R N ) is introverted. To this end, let f ∈ A, and let g lying in the pointwise closure of the orbit of f . There is a net (a ν ) ν ⊂ R N such that τ aν f → g pointwise. But the set {τ aν f : ν} is weakly relatively compact, hence by passing eventually to a subnet, we have τ aν f → g weakly in BUC(R N ). It readily follows that g, viewed as the weak limit of a net in A, belongs to A since A is weakly closed. As a result of Lemma 3, A is introverted.
We are now in a position to establish the structure theorem for the spectrum of an algebra with mean value. This result and its corollaries are the basic results on topological semigroups for the applications made in the following sections.

Theorem 2. Let A be an algebra with mean value on R N . Assume that A is introverted. Then its spectrum ∆(A) is a compact topological semigroup. Moreover if the multiplication in ∆(A) is jointly continuous then ∆(A) is a compact topological group.
Proof. 1. Let us first define the Arens product (see e.g. [7,16]) on ∆(A). For f ∈ A and x ∈ R N , we know that the translate τ x f (defined by τ x f (y) = f (x + y) The operator T ν is nonnegative, sends the constant function 1 into itself and, since A is introverted, maps A into A. We therefore find that, if µ ∈ ∆(A), the product µν determined by is well defined as and element of ∆(A). That it is associative is straightforward. Moreover denoting by δ the Dirac mass at the origin 0, it holds that δν = νδ = ν for all ν ∈ ∆(A), so that δ is a unity in ∆(A) for our product. Finally, for the separate continuity, fix ν in ∆(A) and let us check that µ → µν is continuous. Before we can do this, let us however observe that µν is actually the convolution between µ and ν defined by With this in mind, we follow some arguments of [30] (see in particular Proposition 3.1 therein). Let (µ i ) i∈I be a net in ∆(A) converging weak * to µ in ∆(A). Then the tensor product (µ Hence by the definition of product measure we must have λ = µ ⊗ ν. This shows that lim i∈I (µ i ⊗ ν) = µ ⊗ ν. Since the map α ⊗ β → α * β of ∆(A) × ∆(A) into ∆(A) is continuous (by definition of convolution) we finally have lim i∈I µ i ν = µν. The continuity of µ → µν follows thereby.
The fact that µ → νµ is continuous is a consequence of the commutativity of the product as seen below (see e.g. [11, Theorem 3.1]): 2. Now assume that the multiplication is jointly continuous. Taking into account the first part of the proof, we see that it is sufficient to check that any µ ∈ ∆(A) is invertible. It is clear that if µ = δ y for some y ∈ R N , then ν = δ −y is the inverse of µ, since µν = νµ = δ. Now, assume that µ is arbitrary, and let (y i ) i∈I ⊂ R N be a net such that δ yi → µ in ∆(A)-weak * . (δ −yi ) i∈I is a net in ∆(A), hence possesses a weak * cluster point ν ∈ ∆(A) which is the limit of a subnet (δ −y i(j) ) i∈J of (δ −yi ) i∈I . Invoking both the continuity of the multiplication and the identity δ −y i(j) δ y i(j) = δ, we are led (after passing to the limit) to µν = δ. The uniqueness of ν is a consequence of the commutativity of the multiplication. Thus ∆(A) is a group. Since a compact topological semigroup that is a group must be a topological group (see [8, Theorem 2.1]), it readily follows that ∆(A) is a compact topological group. This concludes the proof.
Let A be an introverted algebra with mean value. Then its spectrum is a compact topological semigroup. We may therefore define the translation operator on C(∆(A)) as follows. For f ∈ C(∆(A)) and r ∈ ∆(A), τ r f (s) = f (sr) (s ∈ ∆(A)). The following holds true.

Corollary 2. Let
A be an introverted algebra with mean value on R N . The mapping ϕ : R N → ∆(A) defined by ϕ(y) = δ y is a continuous homomorphism with dense range. Moreover it holds that where · denotes the Gelfand transformation on A.
Proof. For the continuity of ϕ, let (y n ) n ⊂ R N be a net such that y n → y; then for any u ∈ A, u(y n ) → u(y), hence δ yn → δ y in the weak * topology of A ′ . This proves the continuity of ϕ. The fact that ϕ is a homomorphism just comes from the obvious equality δ x+y = δ x * δ y ≡ δ x δ y for all x, y ∈ R N . Let us check (3.1). Let y, z ∈ R N and u ∈ A; then Using the continuity of ϕ and the density of The following result is of independent interest. It characterizes in terms of the weakly almost periodic functions, the algebras with mean value that are introverted.
thus for each u ∈ A, {τ y u : y ∈ R N } is the continuous image of the set {τ ϕ(y) u : y ∈ R N }. Therefore, in order to prove the theorem, it suffices to check (i) and (ii) Let us now verify (i) and (ii) hold in C(∆(A)).
(i) We use a result of Grothendieck [13] stating that weak compactness and compactness in the topology of pointwise convergence agree on bounded subsets of C(X) for X compact. Bearing this in mind, let f ∈ C(∆(A)). Invoking the separate continuity of the multiplication in ∆(A), we get that the mapping s → τ s f from ∆(A) into C(∆(A)) is continuous when C(∆(A)) is taken in the topology of pointwise convergence. Thus O(f ) is compact in that topology as the continuous image of the compact set ∆(A). We infer from the above mentioned result of Grothendieck that O(f ) is weakly compact.
(ii) It is sufficient to show that the mapping s → τ s f is strongly continuous, for if it is strongly continuous, then O(f ) will be strongly compact as the strongly continuous image of ∆(A). So, let us check the strong continuity of that mapping. The function ∆(A) × ∆(A) → C defined by (s, r) → f (sr) is continuous. Thus, given ε > 0, for a fixed r 0 ∈ ∆(A) and for any s ∈ ∆(A), there is a neighborhood This shows the continuity at r 0 .

Corollary 3. Let A be an algebra with mean value. Then ∆(A) is a group if and only if A is a closed subalgebra of the almost periodic functions.
Proof. It is classically known that if A is a closed subalgebra of the algebra of almost periodic functions, then ∆(A) is a topological group. Conversely, assuming ∆(A) to be a group, the multiplication is jointly continuous and so, by part (ii) of Theorem 3, A is a closed subalgebra of the algebra of almost periodic functions.
, the algebra of those function in BUC(R N ) that converge at infinity. Then the multiplication in ∆(A) is not jointly continuous. Indeed, if we denote by C 0 (R N ) the Banach space of functions vanishing at infinity, then C 0 (R N ) ⊂ A. Now, let y n = (n, ..., n) ∈ N N (N the nonnegative integers). Take µ n = δ yn the Dirac mass at y n , and define µ −n = δ −yn . We have, for any f ∈ C 0 (R N ), µ n (f ) = f (y n ) → 0 and µ −n (f ) → 0 as n → ∞, while µ n µ −n (f ) = f (0). Assuming the multiplication to be continuous, we must have f (0) = 0 for each f ∈ C 0 (R N ), which is not true.
Remark 3. By virtue of [part (i) of] Theorem 3, the introversion of the algebra with mean value A entails its ergodicity since any subalgebra of the algebra of weakly almost periodic functions is ergodic; see e.g. [27,22]. We assume here that N = 1. Let A be the algebra generated by the function f (y) = cos 3 √ y (y ∈ R) and all its translates f (· + a), a ∈ R. It is known that A is an algebra with mean value which is not ergodic; see [14, p. 243] for details. It follows from Theorem 3 that A is not introverted.
Let A be an introverted algebra with mean value. For any s ∈ ∆(A), s∆(A) is an ideal of ∆(A) in the sense that for any r ∈ ∆(A) and each µ ∈ ∆(A), r(sµ) ∈ ∆(A) since r(sµ) = srµ. Now, set The following result provides us with the structure of K(∆(A)). Proof. (i) Set I = K(∆(A)). For s ∈ ∆(A), we have that sI is an ideal contained in I, so sI = I. Thus if s ∈ I, we have an element e in I such that se = s. We infer that rse = rs for any r ∈ I. Since sI = Is = I, it follows that e is an identity for I. On the other hand, it follows from the equality rI = I (for fixed r in I) that there is a µ ∈ I for which rµ = e, and hence, I is a group. But I is compact as it is closed in the compact space ∆(A). Whence I is a compact topological group.
We shall henceforth consider the mean value as an integral over the kernel K(∆(A)) of ∆(A) whenever the algebra A is introverted. We now have in hands all the ingredients necessary to the definition of the convolution product on the spectrum ∆(A) of any introverted algebra with mean value A. To do this, let p, q, m ≥ 1 be real numbers satisfying 1 and v ∈ L q (∆(A)) we define the convolution product u * v as follows: Then * is well defined since K(∆(A)) is an ideal of ∆(A). Indeed for s ∈ ∆(A) and r ∈ K(∆(A)), r −1 exists in K(∆(A)) and sr −1 ∈ K(∆(A)). from the definition of * it holds that u * v ∈ L m (∆(A)) and the following Young inequality holds true: Now let u ∈ L p (R N ; L p (∆(A))) and v ∈ L q (R N ; L q (∆(A))). We define the double convolution u * * v as follows: Then * * is well defined as an element of L m (R N × ∆(A)) and satisfies It is to be noted that if u ∈ L p (Ω; L p (∆(A))) where Ω is an open subset of R N , and v ∈ L q (R N ; L q (∆(A))), we may still define u * * v by viewing u as defined in the whole of R N × ∆(A); it suffices to take the extension by zero of u outside Ω × ∆(A).
Finally, for u ∈ L p (R N ; B p A ) and v ∈ L q (R N ; B q A ) we define the double convolution denoted by * * as follows: u * * v is that element of L m (R N ; B m A ) defined by It also satisfies the Young inequality. For the next result, we need to define a dynamical system on ∆(A). We equip ∆ into ∆(A). The next result will be of a very first importance in the following sections. It is new and constitutes the cornerstone of the connection between the convolution and the Σ-convergence method.
Theorem 5. Let A be an algebra with mean value on R N . Suppose that A is introverted. Then denoting by δ y the Dirac mass at y, it holds that δ y ∈ K(∆(A)) for almost all y ∈ R N .
The proof of this result heavily relies on the following lemma whose proof can be found in [14]. Then there exists a set of full measure Ω 1 ⊂ Ω 0 such that for a given s ∈ Ω 1 , we have T (x)s ∈ Ω 0 for almost all x ∈ R N .

On a De Rham type result
In this section, we assume A to be an algebra with mean value on R N as defined in Section 2. Let u ∈ A and let ϕ ∈ C ∞ 0 (R N ) = D(R N ). Since u and ϕ are uniformly continuous and A is translation invariant, we have u * ϕ ∈ A (see the proof of Proposition 2.3 in [39]), where here * stands for the usual convolution operator. More precisely, u * ϕ ∈ A ∞ since D α y (u * ϕ) = u * D α y ϕ for any α ∈ N N . For 1 ≤ p < ∞, let u ∈ B p A and η > 0, and choose v ∈ A such that u − v p < η/( ϕ L 1 (R N ) + 1). Using Young's inequality, we have We may therefore define the convolution between B p A and C ∞ 0 (R N ). Indeed, for u = u+N ∈ B p A (R N ) (with u ∈ B p A ) and ϕ ∈ C ∞ 0 (R N ), we define u ⊛ ϕ as follows u ⊛ ϕ := u * ϕ + N ≡ ̺(u * ϕ). (4.1) Indeed, this is well defined as justified below by (4.5). Thus, for u ∈ B p A and where Suppϕ stands for the support of ϕ and |Suppϕ| its Lebesgue measure. Indeed, we have where u = G(u) and τ y u = u(· + y). It is easily seen that u ⊛ ϕ ∈ C(∆(A)). We have u * ϕ = u ⊛ ϕ for all u ∈ A and ϕ ∈ C ∞ 0 (R N ). (4.5) Indeed, for x ∈ R N , we have and (4.5) follows by the continuity of both u ⊛ ϕ and u * ϕ, and the denseness of {δ x : x ∈ R N } in ∆(A). As claimed above, (4.5) justifies that u ⊛ ϕ is welldefined by (4.1) for u ∈ B p A (R N ) and ϕ ∈ C ∞ 0 (R N ). Indeed, for u, v ∈ u, we have u, v ∈ B p A with u = v and so u = u + N = v + N . It emerges from (4.5) that We also have the obvious equality The following De Rham type result holds.
Moreover f is unique modulo I p A , that is, up to an additive function g ∈ B p A verifying ∇ y g = 0.
Moreover if div y ϕ = 0 then div y (̺(u * ϕ)) = ̺(u * div y ϕ) = 0, hence L u (ϕ) = 0, that is, L u vanishes on the kernel of the divergence in D(R N ) N . By the De Rham theorem, there exists a distribution S(u) ∈ D ′ (R N ) such that L u = ∇ y S(u). This defines an operator satisfying the following properties: (i) S(τ y u) = τ y S(u) for all y ∈ R N and all u ∈ A ∞ ; (ii) S maps linearly and continuously A ∞ into L p ′ loc (R N ); (iii) There is a positive constant C r (that is locally bounded as a function of r) such that The property (i) easily comes from the obvious equality

Let us check (ii) and (iii). For that, let
the last inequality being due to (4.3). Hence, as Suppϕ i ⊂ B r (1 ≤ i ≤ N ), Now, let g ∈ C ∞ 0 (B r ) with Br gdy = 0; then by [24,Lemma 3.15] there exists ϕ ∈ C ∞ 0 (B r ) N such that div ϕ = g and ϕ W 1,p (Br ) N ≤ C(p, B r ) g L p (Br) . We have and by a density argument, we get that S(u) ∈ (L p (B r )/R) ′ = L p ′ (B r )/R for any r > 0, where L p ′ (B r )/R = {ψ ∈ L p ′ (B r ) : Br ψdy = 0}. The properties (ii) and (iii) therefore follow from the above series of inequalities. Taking (ii) as granted it comes that We claim that S(u) ∈ C ∞ (R N ) for all u ∈ A ∞ . Indeed let e i = (δ ij ) 1≤j≤N (δ ij the Kronecker delta). Then owing to (i) and (iii) above, we have Hence, passing to the limit as t → 0 above leads us to Repeating the same process we end up with So all the weak derivative of S(u) of any order belong to L p ′ loc (R N ). Our claim is therefore a consequence of [28,Theorem XIX,p. 191].
This being so, we derive from the mean value theorem the existence of ξ ∈ B r such that On the other hand, the map u → S(u)(0) is a linear functional on A ∞ , and by the above equality we get Hence, defining S : We infer from both the density of D A (R N ) in B p ′ A and (4.9) the existence of a function f ∈ B p A with f p ≤ c L such that In particular where u = G(u) = G 1 (̺(u)). Now, let u ∈ A ∞ and let y ∈ R N . By (i) we have S(u)(y) = S(τ y u)(0) =

Thus
We deduce that L and ∇ y f agree on (B 1,p ′ A ) N , i.e., L = ∇ y f . For the uniqueness, let f 1 and f 2 in B p A be such that L = ∇ y f 1 = ∇ y f 2 , then ∇ y (f 1 − f 2 ) = 0, which means that f 1 − f 2 ∈ I p A . The preceding result together with its proof are still valid mutatis mutandis when the function spaces are complex-valued. In that case, we only require the algebra A to be closed under complex conjugation (u ∈ A whenever u ∈ A). As a result of the preceding theorem, we have the Then there exists a function u ∈ B 1,p A , uniquely determined modulo I p A , such that f = ∇ y u.
and it follows from Theorem 6 the existence of u ∈ B p A such that f = ∇ y u. This shows at once that u ∈ B 1,p A . The uniqueness is shown as in Theorem 6. Remark 4. Let u ∈ B p A be such that ∇ y u = 0; then u ∈ I p A . Thus, for the uniqueness argument, we may choose the function u in Corollary 4 to belong to B 1,p A /I p A , which space we shall henceforth equip with the norm gradient norm as above; see (2.3).

Sigma convergence method
Throughout this section, Ω is an open subset of R N , and unless otherwise specified, A is an algebra with mean value on R N . ·)) a.e. in x ∈ Ω. We denote this by u ε → u 0 in L p (Ω)-weak Σ.
We recall here that u 0 = G 1 •u 0 and ψ = G•ψ, G 1 being the isometric isomorphism sending B p A onto L p (∆(A)) and G, the Gelfand transformation on A. In the sequel the letter E will throughout denote a fundamental sequence, that is, any ordinary sequence E = (ε n ) n (integers n ≥ 0) with 0 < ε n ≤ 1 and ε n → 0 as n → ∞. The following result holds.
Below is one fundamental result involving the gradient of sequences.
Theorem 8. Let 1 < p < ∞. Let (u ε ) ε∈E be a bounded sequence in W 1,p (Ω). Then there exist a subsequence E ′ of E, and a couple (u 0 , u 1 ) ∈ W 1,p (Ω; Proof. Since the sequences (u ε ) ε∈E and (∇u ε ) ε∈E are bounded respectively in L p (Ω) and in L p (Ω) N , there exist a subsequence E ′ of E and u 0 ∈ L p (Ω; This shows that ∇ y u 0 = 0, which means that u 0 (x, ·) ∈ I p A (see (2.2)), that is, where ψ ε j (x) = ψ j (x/ε). Passing to the limit in the above equation when and reminding that M (v j ) ∈ L p (Ω) we have by (5.3) that ∂u0 ∂xj ∈ L p (Ω; I p A ), where ∂u0 ∂xj is the distributional derivative of u 0 with respect to x j . We deduce that u 0 ∈ W 1,p (Ω; I p A ). Coming back to (5.2) we get and so, as ϕ is arbitrarily fixed,
Remark 5. If we assume the algebra A to be ergodic, then I p A consists of constant functions, so that the function u 0 in Theorem 8 does not depend on y, that is, u 0 ∈ W 1,p (Ω). We thus recover the already known result proved in [27] in the case of ergodic algebras.

On the interplay between Σ-convergence and convolution
In order to take full advantage the results of Section 3, we assume throughout this section that the algebra A is introverted. Then its spectrum is a compact topological semigroup.
With this in mind, let a ∈ R N be fixed. Appealing to Theorem 5, we may assume without lost of generality that (δ a/ε ) ε>0 is a net in the compact group K(∆(A)), hence it possesses a weak * cluster point r ∈ K(∆(A)). In the sequel we shall consider a subnet still denoted by (δ a/ε ) ε>0 (if there is no danger of confusion) that converges to r in the relative topology of ∆(A), i.e. δ a ε → r in K(∆(A))-weak * as ε → 0. (6.1) Finally let Ω be an open subset of R N , and let (u ε ) ε>0 be a sequence in L p (Ω) (1 ≤ p < ∞) which is weakly Σ-convergent to u 0 ∈ L p (Ω; B p A ). We will see that a micro-translation of the sequence u ε induces a micro-translation on its limit, while a macro-translation of u ε induces both micro-and macro-translations on the limit u 0 . This is the main goal of the following result.

Proposition 3.
(i) If p > 1 then Proof. We split the proof into two parts. Part 1. We consider first the case of weak Σ-convergence. Let us first check (6.2). Prior to this, let us note that the proof of (6.2) in the special case p = 1 has been done in [27]. In the general situation when p > 1, the proof is very similar to the one of the previous case, and for that reason, we just sketch it here. Let ϕ ∈ C ∞ 0 (Ω) and let ψ ∈ A; we have As in [27], we have that We infer from the inequality ((Ω + εa)∆Ω being denoting the symmetric difference between Ω + εa and Ω) that (II) and (III) tend to 0 as ε → 0. This proves (6.2). The proof of (6.3) is more involved. It has just been done in [31] (in the case of weak Σ-convergence of course!) under the restricted assumption that Ω is bounded and ∆(A) is a group. We present here a general proof in which all these assumptions are relaxed.
As for (I), we easily verify (using the fact that G is an isometry) that Since the mapping s → s −1 is continuous in K(∆(A)), we get that δ − a ε = δ −1 a ε → r −1 in K(∆(A)) weak * and δ −yn = δ −1 yn → r −1 in K(∆(A)) weak * . Invoking the uniform continuity of ψ, we are led to → 0 as ε → 0 and next n → ∞. Part 2. We now assume strong Σ-convergence of (u ε ) ε . Then from Part 1, we have (6.2) and (6.3) in the weak sense. Moreover we deduce from both the equality u ε L p (Ω) = u ε (· + a) L p (Ω−a) and the translation invariance of the measure β that u ε (· + a) L p (Ω−a) → u 0 (· + a, ·r) L p ((Ω−a)×K(∆(A))) . This concludes the proof of (6.3) in both cases (weak and strong Σ-convergence). The same also holds for (6.2) in the case of strong Σ-convergence. The proof is complete.
The next important result deals with the convergence of convolution sequences. Let p, q, m ≥ 1 be real numbers such that 1 p + 1 q = 1 + 1 m . Let (u ε ) ε>0 ⊂ L p (Ω) and (v ε ) ε>0 ⊂ L q (R N ) be two sequences. One may view u ε as defined in the whole R N by taking its extension by zero outside Ω. Define which lies in L m (R N ) and satisfies the Young's inequality We have the following result.
Theorem 10. Let (u ε ) ε>0 and (v ε ) ε>0 be as above. Assume that, as ε → 0, Proof. In view of (6.4), the sequence (u ε * v ε ) ε>0 is bounded in L m (Ω). Now, let η > 0 and let ψ 0 ∈ K(R N ; A) (the space of continuous functions from R N into For f ∈ K(Ω; A), we have (by still denoting by u ε the zero extension of u ε off Ω) On the one hand one has ( where c is a positive constant independent of ε. It follows from (6.5) that On the other hand, in view of Theorem 9, we have, as ε → 0, where r = lim δ t/ε (for a suitable subsequence of ε → 0) in K(∆(A))-weak * . So let Φ : R N × ∆(A) → R be defined by Then we easily check that Φ ∈ K(R N ; C(∆(A))), so that there is a function Ψ ∈ K(R N ; A) with Φ = G • Ψ. We can therefore define the trace Ψ ε (t) = Ψ(t, t/ε) (t ∈ R N ) and we have As far as (II 1 ) is concerned, set Then the following claims hold.
Claim 1. For a.e. t, V ε (t) → 0 as ε → 0 (possibly up to a subsequence) Indeed, for Claim 1, applying Theorem 9 leads one to where r is such that δ t/ε → r in K(∆(A)) weak * for some subsequence of ε. Moreover, since Ψ ε (t) = Φ(t, δ t/ε ), we have by the continuity of Φ(t, ·) that, for the same subsequence, Whence Claim 1 is justified. As for Claim 2, first and foremost we have where c is a positive constant independent of t and ε. Since f and ψ 0 belong to K(R N ; A) we have that f ε and ψ ε 0 lie in K(R N ) and their support are contained in a fixed compact set of R N . Therefore ψ ε 0 * f ε ∈ K(R N ). As a result, V ε ∈ K(R N ) and further its support is contained in a fixed compact set K ⊂ R N independent of ε.
This being so, let γ > 0. From Egorov's theorem there exists D ⊂ R N such that meas(R N \D) < γ and V ε converges uniformly to 0 on D. We have the following series of inequalities where C 1 is a positive constant independent of both ε and D. It emerges from the uniform continuity of V ε in D that there exists ε 0 > 0 with α 1 ≤ α such that where C 2 > 0 is independent of ε. This shows Claim 2. Thus (II 1 ) → 0 as ε → 0. As for (II 2 ), using once again the weak Σ-convergence of (u ε ) ε , we get u 0 (t, r) Ψ(t, r)dtdβ, Thus, there is 0 < α 2 ≤ α 1 such that Now, let 0 < ε ≤ α 2 be fixed. Finally the decomposition associated to (6.5)-(6.7) allow one to see that Here C is a positive constant independent of ε. This concludes the proof.
Remark 6. Theorem 10 generalizes its homologue Theorem 2 in [31] which is concerned with the special case when q = 1 and Ω bounded. Though the above proof is very similar to the one in [31], it is different from the latter in the sense that it takes full advantage of Egorov's theorem.
In practise, we deal in this work with the evolutionary version of the concept of Σ-convergence. This requires some further notion such as the one related to the product of algebras with mean value. Let A 1 (resp. A 2 ) be an algebra with mean value on R m1 (resp. R m2 ). We define their product denoted by A 1 ⊙ A 2 as the closure in BUC(R m1 × R m2 ) of the tensor product A 1 ⊗ A 2 = { finite u i1 ⊗ u i2 : u ij ∈ A j , j = 1, 2}. It is a well known fact that A 1 ⊙ A 2 is an algebra with mean value on R m1 × R m2 ; see e.g. [21,22].
With this in mind, let A = A y ⊙A τ where A y (resp. A τ ) is an algebra with mean value on R N y (resp. R τ ). The same letter G will denote the Gelfand transformation on A y , A τ and A, as well. Points in ∆(A y ) (resp. ∆(A τ )) are denoted by s (resp. s 0 ). The compact space ∆(A y ) (resp. ∆(A τ )) is equipped with the M -measure β y (resp. β τ ), for A y (resp. A τ ). We have ∆(A) = ∆(A y )×∆(A τ ) (Cartesian product) and the M -measure for A, with which ∆(A) is equipped, is precisely the product measure β = β y ⊗ β τ (see [21]). Finally, let 0 < T < ∞. We set Q T = Ω × (0, T ), an open cylinder in R N +1 .
This being so, a sequence ( Remark 7. The conclusions of Theorems 7 and 8 are still valid mutatis mutandis in the present context (change there Ω into Q T , W 1,p (Ω) into L p (0, T ; W 1,p (Ω)), Now, assume that A y and A τ are introverted. Let (a, τ ) ∈ R N × R, and let the micro-translations r and r 0 being determined as follows: δ a/ε → r in K(∆(A y )) and δ τ /ε → r 0 in K(∆(A τ )) up to a subsequence of ε. A similar conclusion holds in Theorem 10 mutatis mutandis.

Homogenization of a parameterized Wilson-Cowan type equation with finite and infinite delays
We consider the parameterized Wilson-Cowan model with delay [36,37] ∂uε ∂t (x, t) = −u ε (x + a, t) is fixed, u ε denotes the electrical activity level field, f the firing rate function and K ε = K ε (x) = K(x, x/ε) the connectivity kernel. We assume that K ∈ K(R N ; A) (A an introverted algebra with mean value on R N ) is nonnegative and is such that R N K ε (x)dx ≤ 1, f : R N × R → R is a nonnegative Carathéodory function satisfying the following conditions: (H1) For almost all y ∈ R N , the function f (y, ·) : λ → f (y, λ) is continuous; for all λ ∈ R, the function f (·, λ) : y → f (y, λ) is measurable and f (·, 0) lies in L 1 (R N ) ∩ L 2 (R N ); there exists a positive constant k 1 such that |f (y, µ 1 ) − f (y, µ 2 )| ≤ k 1 |µ 1 − µ 2 | for all y ∈ R N and all µ 1 , µ 2 ∈ R.
(H3) For any sequence Assumption (H1) is used to derive the existence and uniqueness result while assumptions (H2) and (H3) are the cornerstone in the homogenization process. It is worth noticing that assumption (H3) is meaningful. It concerns the convergence of fluxes to flux. Indeed the convergence result However, under some circumstances, this becomes true; see e.g. [31,Section 4] for some concrete examples.
In [31], Eq. (7.1) with a = 0 has been considered. Here we aim at showing how the translation induces a memory effect at the microscopic level. Before we can do that, however, we need to provide an existence result. But by repeating the proof of Theorem 3 in [31], we get the following result.
where C is a positive constant depending only on u 0 and on T . Moreover the sequence (u ε ) 0<ε≤1 is uniformly integrable in L 1 (R N T ). In this section, we use the following version of Σ-convergence. A sequence x, t, ·)) a.e. in (x, t) ∈ R N T . With this in mind, we can now state and prove the homogenization result.
Theorem 12. Let (u ε ) ε>0 be the sequence of solutions to (7.1). Then there exists a subsequence of ε still denoted by ε such that, when ε → 0, it holds that δ a ε → r in K(∆(A)) weak * (7.3) and where u 0 ∈ C([0, T ]; L 1 (R N ×∆(A))) is the unique solution to the following equation Proof. Let E be an ordinary sequence of positive real numbers ε. We know from Theorem 11 that the sequence (u ε ) ε∈E is uniformly integrable in L 1 (R N T ). As a result of [part (ii) of] Theorem 7, there exist a subsequence E ′ of E and a function v 0 ∈ L 1 (R N T ; B 1 A ) such that, as E ′ ∋ ε → 0, we have (7.4) with u 0 = G 1 • v 0 . On the other hand, in view of Theorem 5, (δ a/ε ) ε∈E ′ is a sequence in the compact space K(∆(A)), so that there exists a subsequence of E ′ not relabeled and r ∈ K(∆(A)) such that (7.3) holds true. The theorem will be proved once we will check that u 0 ∈ C([0, T ]; L 1 (R N × ∆(A))) and satisfies (7.5). In order to do that, first recall that K ∈ K(R N × (0, T ); A) since K(R N ; A) ⊂ K(R N × (0, T ); A), so that we have On the other hand, using the assumption (H3) in conjunction with (7.4), we are led to We infer from Theorem 10 that with r determined by (7.3); this is a consequence of [part (ii) of] Theorem 9. Finally, noting that Eq. (7.1) is equivalent to the following integral equation we are led (after passing to the limit when E ′ ∋ ε → 0 and using Fubini and Lebesgue dominated convergence results in the integral term) to Therefore, composing both members of the above equality by G 1 , we end up with which is nothing else but the integral form of (7.5). We also conclude from the preceding equation that u 0 lies in C([0, T ]; L 1 (R N × ∆(A))) as expected. Whence the proof.
It is well known that the functions a ε (·, ·, Dv) and a ε 0 (·, ·, Dv) (for fixed v ∈ L 2 (0, T ; W 1,2 0 (Ω))), K ε and ρ ε are well defined as elements of L 2 (Q T ) N , L 2 (Q T ) N , L 1 (Q T ) and L ∞ (Ω) respectively and satisfy properties of the same type as in A1-A2. Finally, choose f in L 2 (Q T ) and u 0 ∈ L 2 (Ω). Our first objective in this section is to provide an existence and uniqueness result for problem (8.1).
≤ K ε * (a ε 0 (·, ·, ∇u 1 ) − a ε 0 (·, ·, ∇u 2 )) L 2 (Qt) u 1 − u 2 L 2 (Qt) . But, in view of Poincaré's inequality, there is a positive constant α depending only on Ω such that It readily holds that Taking the supremum over [0, T ] we end up with Consequently, by shrinking T > 0 in such a way that C √ T < 1, it emerges that the above mapping is contractive. The rest of the proof follows by mere routine.
Remark 8. It follows from the inequalities (8.5) that the sequence (u ε ) ε>0 determined above is relatively compact in the space L 2 (Q T ).
Throughout the rest of this section, A y and A τ are algebras with mean value on R N y and R τ respectively. We set A = A y ⊙ A τ and further we assume that A τ is introverted. It is worth noting that property (5.1) is still valid for ψ ∈ C(Q T ; y,τ ). Bearing this in mind, let (u ε ) ε>0 be the sequence of solutions to (8.1). Our main objective here amounts to study the asymptotic behaviour as ε → 0, of (u ε ) ε>0 . This will of course arise from the following important assumption: a 0 (·, ·, λ) ∈ B 2 A and a(·, ·, λ) ∈ (B 2 A ) N for any λ ∈ R N , K ∈ B 1 A and ρ ∈ A y with M (ρ) > 0.
The homogenization of problems of type (8.1) has been left opened in [2] in which the authors considered only the linear version of such type of equations. They used the Laplace transform to perform the homogenization process. Our work is therefore the first one in which the homogenization of (8.1) is considered, even in the periodic setting.
Remark 9. The conclusion of the above results also hold true for the mapping a 0 , mutatis mutandis.
Therefore, proceeding exactly as in the proof of Theorem 3.10 in [39] (see also the proof of Theorem 4.1 in [40]), we are readily led to K * * z 0 = K * * a 0 (·, ·, D y u). This concludes the proof of the theorem.
Then substituting u 1 = v(∇u 0 ) in (8.14) and choosing there the special test function v 0 = ϕ ∈ C ∞ 0 (Q T ), we quickly obtain by disintegration, the macroscopic homogenized problem, viz., ρ ∂u0 ∂t − div b(∇u 0 ) + b 0 (∇u 0 ) = f in Q T u 0 (0) = u 0 in Ω. (8.17) By the uniqueness of the solution to (8.11), the existence and the uniqueness of the solution to (8.17) is ensured. We are therefore led to the following Theorem 15. Assume that (8.7) holds. For each ε > 0 let u ε be the unique solution to (8.1). Then as ε → 0, where u 0 is the unique solution to (8.17).