Maximal closed ideals of the Colombeau Algebra of Generalized functions

In this paper we investigate the structure of the set of maximal ideals of G(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}(\Omega )$$\end{document}. The method of investigation passes through the use of the m-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m-$$\end{document}reduction and the ideas are analoguous to those in Gillman and Jerison (Rings of Continuous Functions, N.J. Van Nostrand, Princeton, 1960) for the investigation of maximal ideals of continuous functions on a Hausdorff space K.


Introduction
The algebra of Colombeau generalized functions has been used for non linear P.D.E. (see for example [4,[6][7][8]22,24]) as well as for linear P.D.E. with irregular coefficients [21]. Much work has been done on this field from the point view of Analysis. However the Algebraic properties of this Algebra are still a largely open field of investigation.
In the case of continuous functions (with complex or real values over a compact set K it is known that the set of points are in one-one correspondence with the set of maximal ideals. While in the more general case of the algebra of continuous functions on a Hausdorff regular space A, the set of maximal ideals is in a one to one correspondence with theČech-compactification (β(A)) (see [16]).
It is known that generalized functions G( ) have a double structure: (a) They constitute a sheaf of algebras (see [4,6]) and This leads to the notion of 'localisation' of ideals, the notion of 'support' of an ideal and to the notion of generalized trace of ideals (see [2]).
Trying to obtain a better knowledge of prime and maximal ideals of G( ) we make an extensive use of the results concerning the algebraic and topological properties of the ring of generalized constants [3]. In order to use ideas analogous to those in [16], we first use the m−reduction procedure [27] which gives a canonical surjective mapping of G( ) onto G m ( ), but G m ( ) is a topological algebra on the field of m−reduced generalized constants K m = K/m where m is maximal ideal of K. We now consider a very large class of maximal ideals called regular maximal ideals. The set of regular maximal ideals is proved to contain all closed maximal ideals (closed for the natural Hausdorff topology of G( )).
Our main result is the following: the set of regular maximal ideals is in one to one correspondence with (m, d) where m is a maximal ideal of K and d ∈ γ ( m,c ) where γ ( m,c ) is a special compactification of m,c called the g−compactification of m,c .
In Sect. 1. in order to make the paper almost self contained we give the main definitions and results that we will use.
In Sect. 2. we give some results and examples and counterexamples concerning the properties of ideals of G( ).
In Sect. 3. with the help of the concept of m−reduction we investigate the set of maximal ideals of the algebra G m ( ) and show how to transfer these results by pull back on G( ).
In the appendix we present the technique of compactification with the help of special families of closed sets called compactifying families.

Definitions and main results
In order to make this paper almost self contained and accessible to non specialists we remind the main definition and the main algebraic results. In order to enlighten the functoriality of Colombeau constructions we will follow the presentation given in [26].

Colombeau extension
Definition 1.1 Let (E, (μ n ) n ) be a Hausdorff topological vector space whose uniform structure is given by an increasing sequence (μ n ) n of seminorms.
A net ( f ε ) ε ∈ E ]0,1] is said to be moderate if ∀n ∈ N, ∃a n ∈ R s.t. μ n ( f ε ) = o(ε a n ) The vector space of moderate nets will be denoted E M [E].
1) C = R + iR and if a is an ideal of C then a = (a ∩ R) + i(a ∩ R).
2) If p is a prime ideal of K then its closure p (closure for the sharp topology) is a maximal ideal of K. 3) If in R we define x ≤ y by the fact that there exist respective representatives (x ε ) ε and (y ε ) ε s.t. ∀ ε (x ε ) ≤ (y ε ) then we define an order on R (but not a total order). 4) Let us denote by |x| the class of (|x ε |) ε . If U is an ideal of K, |y| ∈ U and |x| ≤ |y|, then x ∈ U.

Generalized functions
Let us consider the space E = C ∞ ( ) where is an open subset of R d . Let us remind first how we can define the uniform structure of E by an increasing sequence of seminorms. Let ( n ) n be an exhaustive sequence of relatively compact subsets of (i.e. n ⊂⊂ n+1 , n = ). If f ∈ C ∞ ( ), we put The ring of moderate nets of C ∞ ( ) will be denoted E M ( ).
The ideal of negligible nets will be denoted N ( ). The Colombeau extension G[C ∞ ( )] will be denoted G( ) and will be called the algebra of Colombeau generalized functions on . By the functoriality properties of Colombeau constructions, one immediately proves that Proposition 1.4 G( ) is a differential topological algebra on K and all differentiations are continuous mappings.
One defines in an obvious way the restriction of a generalized function on an open subset of . In fact we define a sheaf of topological algebras on K which can be proved to be fine [15].
The support of a generalized function f is the complement of the largest open subset such that f / = 0. Clearly K is trivially embedded in a subring of G( ), the subring of 'constant generalized functions'.
A very important property of G( ) is the fact that there exist embeddings of D ( ) into G( ) : Let φ ∈ S(R n ) (Schwartz functions) be such that φdx = 1 and ∀ α, |α| ≤ 1 By sheaf considerations and partition of unity this mapping can be extended on all D ( ). Such mappings respect derivations and products of C ∞ functions but not the products of continuous functions (this would be impossible as is stated by a well known impossibility Theorem of L. Schwartz). However the situation is not completely hopeless because we have in Colombeau theory some useful 'weak-equalities' (which are not stable under products): (a) If F = [( f ε )] and G = [(g ε )], F and G are said to be associated if (b) F and G are said to be strongly associated (see [21,25]) which we denote Oberguggenberger investigated many cases where, when the product of distributions T , S is defined in some sense, then i φ (T · S) is 'weakly equal' to i φ (T )i φ (S) in the sense of one of the above equivalence relations (see [23]).
The functorial structure of Colombeau constructions allows to define the value of a generalized function f on a point x 0 of as well as the integral of f over a compact set K . Moreover it is clear that the mappings f → f (x 0 ) and f → K f dx are continuous K linear mappings for the respective sharp topologies.
Let us remind an important result proved using Landau inequality in [17].
Thus in this case we have to verify the rapid vanishing only of supremum seminorms and we need not evaluate the behaviour of derivatives.
The definition of negligible nets only by the evaluation of the supremum seminorms on relatively compact subsets allows us to prove easily the following.
Injectivity of the mapping G( ) → G [E k ] is easily deduced by the fact that In fact, by Proposition 1.5, the negligibility of the supremums on compact sets implies the negligibility of the supremums of all derivatives on the same sets. As for density, consider the modified nets δ n,ε = 1 ε nd φ x ε n , where φ is a function whose fourier transform is a C ∞ compactly supported function with value 1 on a neighbourhood of 0. Let us now consider (g ε ) ∈ E M,c (E k+1 ) and h ε = g ε − g ε * δ n,ε .

Generalized points
J.F. Colombeau noticed that there exist non zero elements of G( ) which take zero value on all points of . This led M. Kunzinger and M. Oberguggenberger to define the notions of 'generalized points' and 'compactly supported generalized points' (see [19] The set of equivalence classes is called the set of generalized points and is denoted and the class of such a net (x ε ) is denoted as usually by [(x ε )]. A generalized point x = [(x ε )] is said to be supported in a compact subset K if there exists a representative (x ε ) of x such that ∀ ε x ε ∈ K . The set of compactly supported generalized points is denoted c .
where v is the valuation on K) then c acquires a complete ultrametric uniform structure. The topology thus defined is called the sharp topology of c . Clearly is embedded on a discrete subset of c .
In some sense is the 'Colombeau extension' of . By putting for One can easily verify that the mappings are continuous for the respective sharp topologies.
Clearly ϕ i g i define also an element of G( ) and G is well defined because the above sum is locally finite. One verifies immediately that G · f = 1. The sequence presentation was also used in [12] when authors introduced generalisations of the classical Colombeau constructions.

Sequence model of Colombeau theory
In some circumstances the sequence model allows a better understanding of the main points of a proof.
Such proofs can be translated in the net model in the following way. If the proposition is false there would exist a sequence ( f ε n ) satisfying the negation of the proposition and this would give a counterexample in the sequence model.
We will also use the sequence model to present complicated counterexamples where the translation in the net model would just add useless difficulties of comprehension. Since our purpose here is to allow the reader to grasp the main ideas of our work we will freely pass from one model to the other.
Remark However arguments using cardinality must be treated with great caution because the two models don't have the same cardinal as sets.

m−reduction
In usual Colombeau theory the set of generalized constants is not a field but only a ring (contrary to what happens in the 'non standard' version of Colombeau theory (see [20,23], …).
However there is a shortcut to obtain in the frame of standard analysis algebras where the ring of constants is a field.
This procedure is the 'm−reduction' (see [27]) where m is a maximal ideal of K. Those 'm−reduced' constructions are closely analogous to the non standard version of the theory see [20,23,28] and are a direct rough way to construct something similar to ρ E( ).

Definition 1.12
Let m be a maximal ideal of K. The quotient field K m = K/m will be called the field of m−reduced generalized K-constants. The class of x will be denoted [x] m and if x is represented by x ε , it will be denoted [(x ε )] m .
In [2], the authors proved that any maximal ideal m of K is always closed for the sharp topology of K. Thus K m can be endowed with a quotient Hausdorff topology also called the sharp topology of K m .
More precisely The topology and uniform structure thus defined are called the 'sharp' topology and uniform structure of K m .
K m with its sharp uniform structure satisfies the following properties (see [27]).
Then there exists a unique element x 0 of R such that |x − x 0 | is smaller than every positive standard real number, x 0 will be called the shadow of x and we will say that x belongs to the 'halo' of x 0 . h) Then the halo of zero is a subring of R m , both closed and open for sharp topology.
Here we will remind only the proof of g). As R m is totally ordered, let A be the set of classical reals bigger than x. A is non empty and bounded since x is limited. Let x 0 be the infimum of A. For any n ∈ N, |x − x 0 | < 1 n . Notice that on C m we can define an 'absolute value' with value in R We can now easily give the definition of the m−reduced Colombeau extension of (E, (μ n ) n ) (see [27]).

Definition 1.15
Let (E, (μ n )) be a topological vector space topologized by an increasing sequence of seminorms (μ n ).
If ϕ is a continuously tempered mapping (see [9] and [27]) from E to F, then it defines a continuous mapping from Thus the m−reduction is also a 'Functorial' procedure (see [27]).

m−reduced generalized points
The above 'm−reduction' procedure can also be applied on the case of the set of generalized points: The set of equivalence classes thus obtained is denoted m . The set of classes admitting a compactly supported representative will be denoted m,c and will be called the set of compactly supported m−reduced generalized points. The sharp topology is defined also on m,c by the same procedures as previously.
An important property of m,c (whose counterpart is not valid for c ) is the following.
The set of elements of m,c admitting x 0 as a shadow is called the halo of x 0 .
Proof If m,c ⊂ (R m ) n the sentence can be easily proved using proposition 1.14 g) for any given coordinate thus for the whole space (R m ) n , i.e. one can prove that all limited elements of (R m ) n admit such a shadow and all elements of m,c are limited. More over it is obvious that their shadow lies in .
In some sense m,c is some kind of a 'nonstandard' extension of . As in the case of G m [E] we can define on m,c a 'sharp' topology which coincides with the topology given by the R This topology can be given also by an ultrametric distance (see [27]).

The algebra G m (Ä)
Starting from E = C ∞ ( ) with its set of seminorms μ n we obtain the algebra G m ( ) of m−reduced generalized functions (for details see [27]). In this case the subspace of m−negligible elements will be denoted J m ( ).
Using the same method as M. Kunzinger and M. Oberguggenberger in [19] and the fact that all ideals of K are convex (i.e. if |x| ≤ |y| and |y| belongs to the ideal, so does x) we obtain the following important remark.
We give the proof in the simplified case of one variable. By multiplying f ε by a C ∞ compactly supported generalized function ϕ, we can without loss of generality (evaluation been always done on compact spaces) suppose that f ε ϕ = g ε has moderate uniform evaluations for all derivatives. Let us remind the Landau inequalities : if M 0 is the maximum of a function f , M 1 the maximum of its derivatives and M 2 the maximum of its second derivatives, then ,ε be the corresponding maximums for the function g ε , then we will prove that if But as m is a maximal ideal, we can conclude that also [(M 1,ε )] belongs to m since it is a prime ideal. By induction we can prove that, for all orders, the bounds of derivatives belong to m, which implies that ( f ε ) ∈ J m ( ).
One can easily verify that, as in the case of G( ), we define by this procedure a sheaf of algebras and a notion of restriction to open subsets.
Let ı θ be an embedding of D ( ) into G( ). If for any T ∈ D ( ) we associate its image [ı θ (T )] m into G m ( ), we obtain an embedding of distributions into the algebra G m ( ) respecting derivations of distributions and products of C ∞ functions. But this time the 'constants' of our algebra form a field and not just a ring as in the case of G( ).
Clearly K m is embedded in an obvious way into G m ( ) and its elements can be considered 'constant' generalized m−reduced functions.
As in the case of G( ) we can evaluate m−reduced generalized functions on m−reduced compactly supported generalized points.

does not depend on the choice of representatives ( f ε ) and (x ε ). This m−reduced generalized constant will be called the value of f on x, ( f (x)).
Moreover the mappings f → f (x) and x → f (x) are continuous for the respective sharp topologies.
The proof of this proposition is straightforward (see [25]) and is an easy consequence of the 'functoriality' of m−reduction. One can also easily prove that G m ( ) is a topological differential algebra for a fixed K m where K = R or C.
If φ : C → C is an element of O M (C) and if g ∈ G m ( ), we can define φ • g : ] m does not depend on the choice of representatives. It will be denoted φ(g) or φ • g.
Proof Let ( g ε ) be another representative. For any convex compact subset K of , there exists a temperate mapping ε → R ε ≤ C ε N (N , C, being adequate constants) such that for any ε and any x ∈ K both g ε (x) and g ε (x) belong to B(0, R ε ). Thus As any compact set is included in a finite union of balls, the proof is complete.

Proposition 1.21 Maximum theorem.
If f ∈ G m ( ) and ⊂⊂ is a relatively compact subset of then there exists Now we can easily establish the characterization of m−reduced generalized functions by their values on m−reduced generalized points, which amounts to the following proposition: This is an easy consequence of the previous proposition.
Proof For simplicity without loss of generality we suppose that is convex (going to the general case is straightforward but cumbersome). The hypothesis implies that there exist representing nets ( f 1,ε ) and ( f 2,ε ) of f , (x ε ) representing x and (y ε ) representing y and (c ε ) representing c s.t.
Since |y − x| is a non zero element of K m , it is an invertible element of K m . Thus |y − x| has a positive inverse which is majorated by ε −a , where a is some positive real constant. We can conclude that |y − x| is minorated by ε a which implies that it has an invertible representative. For fixed representatives of x and y, we can freely choose f 1,ε and f 2,ε satisfying the above conditions. Let be a convex relatively compact subset of s.t. (x ε ) and (y ε ) are both supported in it. Consider now a net ( ε ) of affine functions s.t.
When an m−reduced generalized function takes non zero values on all elements of ( m,c ), it turns out that it is invertible (contrarily to what happens in the non m−reduced setting where we have to suppose that all values are invertible).

Proposition 1.24 f ∈ G m ( ) is invertible if and only if
Proof Necessity is obvious. Let us now suppose that ∀x ∈ m,c , f (x) = 0. Let ⊂⊂ be a relatively compact open subset of and ( f ε ) ε be a net representing f . There exists for any ε an element Let us now consider a locally finite covering of by relatively compact open subsets (ω i,i∈I ) and construct h i ∈ G m (ω i ) as above. Consider also a C ∞ − partition of unity (ϕ i ) i∈I adapted to the covering (ω i ) i∈I . Then ϕ i h i has a compact support included in ω i and thus defines also an element of G m ( ). The partition being locally finite we can define H = ϕ i h i ∈ G m ( ). Now we clearly have:

'Localization' of ideals
We saw that the structure of generalized function has two aspects: they are a sheaf of algebras and they define continuous mapping from the set of compactly supported generalized points into the ring of generalized constants. Let us first give a notion linked to the 'sheaf theoretic' aspect of the theory. Let J ⊂ G( ) be an ideal of G( ). If ⊂ is an open subset of it might happen that J / contains all generalized constants in which case it spans G( ), i.e. it is locally 'irrelevant'.
Thus studying an ideal J it is relevant to know 'where' it does not span everything. This leads to the following notion already investigated in [2].  We will give a rapid sketch of the proof (for more details see [2]).

Proof
) and hence no point is irregular. Conversely.
Let us suppose that supp(J ) = ∅. By hypothesis for any x ∈ there exists an open neighborhood ω x and f x ∈ J such that f x / ω x = 1. Let be an open relatively compact subset of .
being compact there exists a finite subcovering (ω a n ) of and a C ∞ partition of unity ϕ n adapted to this subcovering. Clearly ϕ n f n / = 1. Thus J is dense. (c) Let us suppose that card(supp(J )) > 1. Then there would exist two points x 1 and x 2 of , both belonging to supp(J ) and x 1 = x 2 . Let ϕ 1 , ϕ 2 be two elements of G( ) such that supp(ϕ 1 ) ∩ supp(ϕ 2 ) = ∅ and such that there exist neighborhoods ω 1 and ω 2 of x 1 and x 2 respectively such that ϕ 1 /ω 1 = 1 and ϕ 2 /ω 2 = 1. Since ϕ 1 · ϕ 2 = 0 ∈ J , ϕ 1 or ϕ 2 has to be in J (J is prime). Thus x 1 or x 2 has to be regular for J .

Remark
The proof of (b) has the following consequence (see [2]).

Generalized trace of ideals
As generalized functions define continuous mappings of c into K, we can define Thus it is relevant to know on which ξ ∈ c , J (ξ ) is a proper ideal: ∀ξ ∈ A f (ξ ) ∈ a (see [2]).
Generalized traces have the following properties (see [2]): (b) Let be convex. If |x − x 0 | ∈ a, then x ∈ Tr(G x 0 ,a ( )). More generally the trace of G A,a consists of all elements y ∈ c s.t. there exists ξ ∈ A s.t. y − ξ ∈ a.
Proof (a) It is straightforward using the definition of generalized trace. (b) Let K be a convex compact subset of in which both y and ξ are compactly supported then thus there exists a positive generalized constant R s.t. [ One can also easily verify the following (see [2]): If an ideal M of G( ) is a maximal ideal then either it is dense, in which case its support and trace are void, or it is closed, in which case its support is exactly one point of .
One might hope that all closed maximal ideals of G( ) are of the form G ξ,m ( ) where ξ ∈ c and m is a maximal ideal of K.
If all closed maximal ideals were of this form, we would at least have finished with the characterisation of closed maximal ideals of G( ) (they would be in a one to one correspondence with couples (ξ, m) where ξ ∈ c and m maximal ideal of K more or less in the same way that the maximal ideals of C(K ) where K is compact are in correspondence with elements of K ).
Unfortunately this is not the case (see [2]).

Proposition 2.8 There exist closed maximal ideals which are not of the form G ξ,m ( ).
Proof Let x 0 be an element of and J 1 the set of all generalized functions f such that x 0 / ∈ supp( f ). Let J 2 be the ideal spanned by J 1 and the following subset A of G( ) A = {ϕ ∈ G( ) : ∃ξ ∈ c , ∃ p > 0 s.t. there exists (ϕ ε ) representative of ϕ and (ξ ε ) representative of ξ such that for any fixed ε, ϕ ε (ξ ε ) = 1, supp(ϕ ε ) ⊂ B(ξ ε , ε p )}. J 1 is a proper ideal. Let us now prove that J 2 is also a proper ideal of G( ). If this was not the case it would be possible to write 1 = f + n k=0 g k ϕ k where f ∈ J 1 and all ϕ k are elements of A. But as x 0 / ∈ supp( f ), there exists a neighbourhood ω of x 0 s.t. f / ω = 0, thus inside ω we would have 1 = n k=0 g k/ω · ϕ k/ω . But n k=0 g k ϕ k/ω takes non zero values only on a finite union of sets of the form the number of these sets and the p being independent of ε. Since for ε small enough, the measure of this union is less than the measure of ω, this sum cannot take value 1 on all points of ω c .
Consider now M a maximal ideal containing J 2 . We will prove that M cannot be dense. If it was dense its support would be void, thus there would exist a neighbourhood v of x 0 and an element ψ of M s.t. ψ/ v = 1. Let now v ⊂⊂ v be a smaller neighbourhood of x 0 . One can construct a partition of unity (ϕ 1 , ϕ 2 ) on adjusted to the covering (v, c v ) i.e. suppϕ 1 ⊂ v, suppϕ 2 ⊂ c v (where c denotes the complement) and ϕ 1 + ϕ 2 = 1. But by definition ϕ 2 belongs to J 1 and hence to M, hence 1 = ϕ 1 + ϕ 2 = ϕ 1 ψ + ϕ 2 ∈ M which is impossible since M is a proper ideal.
Concerning maximal ideals of G( ) we also have the following property.

Proposition 2.9 If M is a maximal ideal and is stable under locally finite sums, then M is closed.
Proof If M was not closed it would be dense hence M would contain the ideal G c ( ) (see Corollary 2.3). But if {ω n , n ∈ N} is a countable locally finite covering of where ω n are relatively compact open subsets of and ϕ n is a C ∞ partition of unity in adapted to this covering then all (ϕ n ) would belong to G c ( ) and hence to M and 1 would also belong to M since M is supposed to be stable under locally finite sums.

Proposition 2.10 If M is a maximal closed ideal of G( ) then M ∩ K = m is a maximal ideal of K.
Proof M ∩ K is a prime ideal of K. But K is a topological subspace of G( ) (here we identify elements of K with the generalized constant functions they define). Thus M ∩ K is a closed prime ideal of K, hence a maximal ideal (see [2]).
A natural question is the following: is it always true that if M is a maximal ideal (dense or closed) then M ∩ K is a maximal ideal of K?
Unfortunately this is not true as we will see by the following counter example.

Proposition 2.11
There exist dense maximal ideals M of G( ) such that M ∩ K is not a maximal ideal of K.
Proof Our counterexample will be given in G(R d ), the passage to a counterexample in G( ) for a general is straightforward but cumbersome. Also for the convenience of the reader, the example will be presented in the sequence model of the theory. (In order to translate the example into the net model, wherever we refer to a representative ( f n ) n , consider the moderate netf ε defined by ∀ ε ∈] 1 n + 1 , Consider now a disjoint family A n of subsets of N such that each A n is infinite and A n = N.
Let (u n ) be a sequence of open subsets of R d such that u n ⊂ B(0, n + 1) − B(0, n) and for each u i consider a relatively compact open subset u i of u i . Let (ϕ n ) be a countable family of C ∞ functions such that 0 ≤ ϕ n ≤ 1 ϕ n / u n = 1 and supp(ϕ n ) is a compact subset of u n .
We also choose u n , u n and ϕ n in such a way that all derivatives of all ϕ n are bounded by constants independent of n.
Consider now the sequence (g n ) defined by g n := ϕ k when n ∈ A k . Clearly as all A k are infinite the sequence is not negligible and as all derivatives are bounded the sequence is moderate, so g = [(g n )] ∈ G(R d ).
Consider now the following generalized constants (considered as generalized functions on R d ).

e. X k is the class of the characteristic function of the subset A k ).
Let I be the ideal of G(R d ) generated by (1 − g) and the family X k .

Lemma 2.12 1 does not belong to I .
Proof Suppose the contrary and let 1 = N k=1 and when n ∈ A N +1 all X k,n in the above expression are zero and g n = ϕ N +1 which implies that As A N +1 is infinite, the above sequence z n cannot be equivalent to 1 in the quotient space K. Thus 1 does not belong to I . ϕ k X k = g (because X k,n · X k,n = X k,n ).
is the limit of [ N k=0 ( 1 n ) k X k,n ] which belongs to I . Thus belongs to the closure of I . Thus g also belongs to the closure I of I because I is an ideal. But by definition (1 − g) belongs to I and hence to I ; thus 1 = g + (1 − g) belongs to I .
Let now M be a proper maximal ideal containing I . Note now that does not belong to M. Otherwise, 1 would also belong to M and M would not be proper ideal but M does contain all the generalized constants 1 n k X k,n (they belong also to I ⊆ M), but the sequence (A N ) N converges to thus M ∩ K is not closed. This implies that it is not maximal, because all maximal ideals of K are closed.
A natural question is now the following: do there exist dense maximal ideals of G( ) whose constants are a maximal ideal?
The answer is yes, as can be seen by the following example. Let I be the ideal spanned in G( ) by a given maximal ideal m of K and G c ( ). Clearly I is a proper ideal and any maximal ideal M of G( ) containing I would verify M ∩ K = m.
This remark justifies the following definition.

Definition 2.14
In a heuristic sense the problems in the structure of a non regular maximal ideal are concentrated near the border of and such maximal ideals are not relevant for most natural problems in Analysis.

Proposition 2.15 Let m be a maximal ideal of K. Then J m ( ) is the ideal of G( ) spanned by m (the elements of m being considered as constant generalized functions).
Proof Let (ϕ n ) be a partition of unity of , where ϕ n are elements of D( ) for which the supports are a locally finite covering of . Let f be an element of J m ( ), f = ϕ n f . Let h n,k be the derivatives of all orders (including 0) of ϕ n f . As ϕ n f has compact support and f belongs to J m ( ), α n,k = sup |h n,k | belongs to m. For any n, k ∈ N, consider q n,k such that α n,k = [(ε q n,k )]α n,k < [(ε n+k )]. As the valuation of α n,k tends to infinity when n + k tends to infinity, the series α n,k converges to an element α of m (the maximal ideal m being closed). It is clear that there exists g n ∈ G c ( ) such that ϕ n f = αg n with support of g n included in support of ϕ n . Thus f = ϕ n f = αg n = α g n ∈ mG( ). Our next step will be the characterisation of the family of all regular maximal ideals of G( ).
Let us now notice that given a maximal ideal m of K, if π m is the canonical map of G( ) into G m ( ), then π m is continuous and surjective.
Our main result in this paper will be to prove that the family r of all regular ideals of G( ) is in a one to one correspondence with the set of couples (m, ξ) where m is a maximal ideal of K and ξ an element of a special compactification of m,c called the g−compactification of m,c and will be noted γ ( m,c ). This compactification is 'smaller' than theČech compactification of m,c .
We are thus led to the study of ideals and maximal ideals of G m ( ).

Generalities
In the case of G m ( ) the study of ideals is greatly simplified by the fact that the ring of generalized constants is a field and thus {0} is the only proper ideal of the ring K m .
As G m ( ) has both sheaf properties and continuity properties as a subset of continuous maps from m,c into K m we have notions of 'support' and 'generalized trace':

Corollary 3.5 A maximal ideal M of G m ( ) is closed if and only if
card(supp(M)) = 1.
Concerning generalized trace we can have some more precise properties than in the case of G( ).

Proposition 3.6 If a is a prime ideal of G m ( ), then the generalized trace of a has at most one element.
Proof Let us suppose that ξ 1 and ξ 2 are two distinct m−reduced generalized points. This implies that there exist p > 0 and nets (ξ 1,ε ), (ξ 2,ε ) representing respectively ξ 1 and ξ 2 and their distance net not belonging to m, such that for ε small enough ξ 1,ε − ξ 2,ε > ε p . But in this case it is possible to construct two nets ϕ 1,ε and ϕ 2,ε , both belonging to E M ( ) such that supp(ϕ 1,ε ) ∩ supp(ϕ 2,ε ) = ∅ for ε small enough and ϕ i, We clearly have ϕ 1 · ϕ 2 = 0 ∈ a, thus ϕ 1 or ϕ 2 has to belong to a. But ϕ i (ξ i ) = 1 thus ξ 1 or ξ 2 does not belong to the generalized trace of a. The existence of maximal closed ideals of G m ( ) with void trace is proved exactly as in the case of G( ) (the same construction gives a counterexample).
In order to characterize maximal ideals, we will now exhibit important relations between ideals and their 'zero sets'. This will follow the ideas used in [16] when they study ideals of the algebra of real valued continuous functions on a completely regular Hausdorff space.

Generalized zero sets of Ä m,c
Clearly, as m−reduced generalized functions define continuous mappings from m,c into K, it is clear that all generalized zero sets are closed subsets of m,c . A natural question is the following: is any given closed subset of m,c a generalized zero set? The answer is no. The proof is based on the following lemma.

Lemma 3.10 If is connected and A and B are two complementary subsets of m,c , they cannot both be Z g −sets.
Proof Let f be such that Z g ( f ) = A and g such that Z g (g) = B. Clearly Z g ( f ) = Z g (| f | 2 ) and Z g (g) = Z g (|g| 2 ). Notice that This implies that |g| 2 + | f | 2 is invertible (Proposition 1.24). Consider now the reduced generalized function let A be the halo of zero and B be the complement of A. Then one can easily verify that A and B are both closed.

Proposition 3.11 The family of Z g −sets is stable under countable intersection.
Let (Z g ( f n )) be a sequence of Z g −sets. Let (μ n ) be an increasing sequence of seminorms defining the topology of C ∞ ( ). Then we can find positive integers m n such that μ n ([(ε m n )]| f n | 2 ) ≤ [(ε n )], ∀n. Then Hence a countable intersection of generalized zero sets is a generalized zero set, too.

Proposition 3.12 The family of Z g −sets is a base for the family of closed subsets of m,c i.e. every closed subset is an intersection of Z g −sets.
This is an easy consequence of the following lemma:  B(0, 1). Consider the net ϕ ε (x) = ϕ x − ξ ε ε p . Clearly this net is moderate and for ε small enough, its support is included in a relatively compact subset of . Thus we can define [(ϕ ε )] m = f . One can now easily verify that f satisfies the requirements of the lemma.

Z g −sets, Z g −filters
As we already saw, a Z g −set is a subset of m,c s.t. there exists an m− reduced generalized function f s.t. f takes zero values on all elements of this subset and only there. These sets will play a role analogous to z−sets in [16] for the investigation of properties of ideals and the study of the family of maximal ideals.
is a finite family of elements of Z g , then n≤N A n ∈ F. (c) ∅ does not belong to F. Proposition 3.15 Given a Z g −filter F, the family J (F) of all m−reduced generalized functions on whose generalized zero set belong to F is a proper ideal.
If f ∈ J (F) and g ∈ J (F), then This leads to the following definition. There exist ideals which are not of this type: consider the ideal J 1 of G m (R) spanned by x and consider the ideal J 2 spanned by x 2 . Those ideals have the same family of Z g −sets, but they are not the same. Proof Let A = Z g ( f ), and B = Z g (g) where g ∈ J . Let us suppose that A ⊃ B. In this case Z g ( f ) = Z g ( f g), because f g(ξ ) = f (ξ )g(ξ ) = 0 if and only if f (ξ ) is zero or g(ξ ) is zero (K is a field). But as Z g ( f ) ⊃ Z g (g), we see that f (x) is zero if and only if f g(x) is zero. But as f g ∈ J , we have proved that A = Z g ( f ) = Z g ( f g) belongs to F(J ).

Let now A and B belong to F(J
Finally, ∅ / ∈ F(J ) because of Proposition 1.24.
Proof If x 0 / ∈ supp(F), then there exists a ball B(x 0 , ρ) ⊂ s.t. there exists A ⊂ m,c , A ∈ F s.t. B(x 0 , ρ) does not contain any shadow of an element of A. Consider ϕ ∈ D( ) s.t. supp(ϕ) ⊂ B(x 0 , ρ) and ϕ/ B(x 0 , ρ 2 ) = 1. Z g (ϕ) is in the complement of B(x 0 , ρ) (considered as a part of m,c ). Thus Z g (ϕ) ⊃ A, thus Z g (ϕ) ∈ F, thus ϕ ∈ J (F) and ϕ = 1 on a neighbourhood V of x 0 , hence x 0 is regular for J (F). There exists a neighbourhood V of x 0 and f 0 ∈ J (F) s.t. f 0 /V = 1. Thus the zero set of F 0 has a shadow that does not intersect V . An analogous argument holds for regular points of P and F(P).

Definition 3.25
A Z g −filter F is said to be prime if, whenever the union A ∪ B of two Z g −sets A and B belongs to F, then A ∈ F or B ∈ F.
In the case of Z g −ideals, there is a natural relation between the notion of prime ideal and prime Z g −filter.

Proposition 3.26 Let P = J (F). Then P is prime if and only if
The converse argument is of the same kind.

The compact set ( Ä m,c )
Following the ideas of theČech-compactification, we will now see how the family of Z g −ultrafilters constitutes a compact space into which m,c is densely embedded. The main idea will be to show that the family of Z g −sets is a compactifying family for m,c (see Appendix).

Proposition 3.27
The family Z g of Z g −sets satisfies the following properties: (2) For any closed subset F of m,c and x ∈ c F, there exists F ∈ Z g s.t. x / ∈ F and F ⊃ F.
(4) Z g is stable under countable intersections and under finite unions. 5) If x ∈ m,c , then {x} ∈ Z g .
(2) It is exactly the content of Lemma 3.13.
(3) Let us suppose We will use the following lemma: Lemma 3.28 Let f ∈ G m ( ) and let a be a positive real number. Then there exists Clearly φ is a bounded smooth function and φ has compact support, thus g = φ(| f | 2 ) ∈ G m ( ). Clearly g takes value zero on all compactly supported generalized points Consider now h 1 ∈ G m ( ) s.t. for any real η > 0 The existence of h 1 and h 2 can easily be deduced by the previous lemma. Put now F 1 = Z g (h 1 ) and F 2 = Z g (h 2 ). Clearly, (4) By Proposition 3.11 and because Z g ( f ) ∪ Z g (g) = Z g ( f g).  ( m,c )). This is not the case. To prove it, we will use Theorem B.1 of Appendix and thus we only have to prove that the there exists a pair (F 1 , F 2 ) of disjoint closed subsets of m,c such that there does not exist a pair F 1 , F 2 of Z g −sets s.t. F 1 ⊂ F 1 , F 2 ⊂ F 2 and F 1 ∩ F 2 = ∅. This is obvious because we saw that there exist pairs of complementary closed subsets of m,c , but there does not exist a pair of complementary Z g −sets.
We have at last proved the main theorem of this section. By pull back we obtain the following: We have thus classified all regular maximal ideals, but we still have no idea concerning the family of non-regular maximal ideals (i.e., the dense maximal ideals M s.t. M ∩ K is not a maximal ideal of K).
There is also another field of analogous investigations. In [5], the authors investigated the behaviour of objects of the theory under the additional hypothesis that we impose parametrizations by ε to be continuous on ε. Which results of this paper are still valid under this condition? For the case of generalized constants, a first investigation has been done in [18].
(2) For any closed subset F of E and every element x in the complement c F of F there exists F ∈ S s.t. F ⊃ F and x ∈ c F . (3) For any couple F 1 , F 2 of elements of S s.t. F 1 ∩ F 2 = ∅ There exist F 1 , F 2 elements of S s.t.
Thus O satisfies the condition to be the family of open subsets of some topology. Let us now prove that E H is Hausdorff. Let x 1 and x 2 be two distinct H−ultrafilters. Then there exist D 1 ∈ x 1 and D 2 ∈ x 2 s.t. D 1 ∩ D 2 = ∅. (Indeed, if D 1 ∈ x 1 is such that D 1 ∩ D 2 = ∅, for each D 2 ∈ x 2 , then there exists an H−filter generated by x 2 and D 1 . By maximality, this H−filter coincides with x 2 . Thus D 1 ∈ x 2 .) Then by property 3 of Definition A.1, there exist D 1 , D 2 elements of H s.t.
Let F i be the H−filter of all elements of H containing D i . Since D 1 ∪ D 2 = E, a maximal H−filter either contains D 1 or contains D 2 (or both). Thus, using the definitions we showed that  1 , i 2 , . . . , i n ∈ I D i 1 ∈ F i 1 , D i 2 ∈ F i 2 , D i n ∈ F i n s.t. D i 1 ∩ D i 2 ∩ · · · ∩ D i n = ∅ (if it would not, the H−filter generated by all F i would exist and satisfy the non empty intersection property.) The above property implies by definition that n j=1 O F i j = E H .

B Comparison of compactifications
An interesting question is the following: if F and G are two compactifying families such that G ⊂ F, under which conditions do G and F give the same compactification? This is investigated in the following theorem.
Theorem B.1 Let E be a normal topological space and let F and G be two compactifying families such that G ⊂ F. Then G and F yield the same compactification if and only if for any couple (F 1 , F 2 ) of elements of F such that F 1 ∩ F 2 = ∅, there exists a couple (G 1 , G 2 ) of elements of G such that F 1 ⊂ G 1 , F 2 ⊂ G 2 and G 1 ∩ G 2 = ∅.
Proof Let us first prove the necessity: Let (F 1 , F 2 ) be a couple of elements of F such that F 1 ∩ F 2 = ∅ and such that there does not exist a couple (G 1 , G 2 ) of elements of G such that: Let G 1 be the set of all elements of G containing F 1 and let G 2 be the set of all elements of G containing F 2 .
By hypothesis we know that if A ∈ G 1 and B ∈ G 2 then A ∩ B = ∅. Let U be a G−ultrafilter containing all such intersections. Let us first prove that there is no element of U which is disjoint from F 1 (respectively F 2 ).
If there existed h ∈ U s.t. F 1 ∩ h = ∅, then by property 3 of compactifying families there would exist h 1 ∈ G, h 1 ⊃ F 1 such that h 1 ∩ h = ∅. But by hypothesis since h 1 ∈ G, h 1 is also in U and h 1 ∩ h = ∅, which is impossible (by construction of U), thus all elements of U intersect F 1 (respectively F 2 ). Let K 1 be an F−ultrafilter containing F 1 and U and K 2 an F−ultrafilter containing F 2 and U. Since F 1 ∩ F 2 = ∅, K 1 and K 2 are distinct ultrafilters.
Let us now prove that the condition is sufficient: Let U be an ultrafilter of elements of G and K 1 and K 2 be two F ultrafilters both containing U. We will prove K 1 = K 2 .
Let us suppose this is not the case. Then there would exist F 1 ∈ K 1 , F 2 ∈ K 2 such that F 1 ∩ F 2 = ∅ and by hypothesis, there would exist G 1 and G 2 elements of G such that G 1 ⊃ F 1 , G 2 ⊃ F 2 , G 1 ∩ G 2 = ∅ but G 1 ∈ K 1 . Thus G 1 ∈ K 1 ∩ G and G 2 ∈ K 2 ∩ G. Both K 1 ∩ G and K 2 ∩ G are filters containing U, which is an ultrafilter. Thus K 1 ∩ G = U = K 2 ∩ G and G 1 ∩ G 2 = ∅ which is impossible. Thus K 1 = K 2 .
Conversely, let K be an F ultrafilter. Let us consider K ∩ G. K ∩ G is clearly a G−filter.
Let G ∈ G such that G / ∈ K ∩ G. This implies that there exists F ∈ K s.t. G ∩ F = ∅ (K is a F−ultrafilter). Then if G 1 and G 2 are two elements of G such that G 1 ⊃ G and G 2 ⊃ F and G 1 ∩G 2 = ∅, G 2 ∈ K. Thus G ∩G 2 = ∅ with G 2 ∈ G ∩K. Thus K ∩ G is a G−ultrafilter. We thus have a one to one correspondence between G−ultrafilters and F−ultrafilters. The continuity of this correspondence comes from the density of E into both compactifications.
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