Extensions of hermitian linear functionals

We study, from a quite general point of view, the family of all extensions of a positive hermitian linear functional ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}, defined on a dense *-subalgebra A0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {A}}_0$$\end{document} of a topological *-algebra A[τ]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {A}}[\tau ]$$\end{document}, with the aim of finding extensions that behave regularly. The sole constraint the extensions we are dealing with are required to satisfy is that their domain is a subspace 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}$$\overline{G(\omega )}$$\end{document}, the closure of the graph of ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} (these are the so-called slight extensions). The main results are two. The first is having characterized those elements of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {A}}$$\end{document} for which we can find a positive hermitian slight extension of ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}, giving the range of the possible values that the extension may assume on these elements; the second one is proving the existence of maximal positive hermitian slight extensions. We show as it is possible to apply these results in several contexts: Riemann integral, Infinite sums, and Dirac Delta.


Introduction
Let be a topological *-algebra (in general, without unit), with topology and continuous involution * , and let 0 be a dense *-subalgebra of . Given a positive hermitian linear functional on 0 (see Definition 2.1 below) is it possible to extend to some elements of ? In this paper we will continue the analysis undertaken in  with the aim of finding extensions that behave regularly.
The previous problem may have, in some situations, easy solutions, namely when is -continuous or closable [3,22] (see Definition below).
In 2010 Bongiorno and two of us (CT, ST) proposed [3] to use the notion of slight extension given in [10, ch.7 §36.7] for nonclosable linear maps for studying extensions of linear functionals, moving from the basic example of the Riemann integral regarded as a linear functional on 0 , the *-algebra C(I) of complex valued continuous functions on a compact interval I ⊆ ℝ , considered as a dense *-subalgebra of , the *-algebra of Lebesgue measurable functions on I with the topology of the convergence in measure. The involution * is given by pointwise complex conjugation. As noted in [3], it is quite elementary to realize that R is not closable and this fact is responsible of the existence of several procedures for extending the integral that, starting from the Lebesgue integral, cover an extensive literature.
Coming back to the general case, three important questions arise. The first is for which elements of it is possible to find a slight extension which is still positive hermitian. The second is, given a such element, what values can the extension assume on it. The third is whether, for each choice, we can find a maximal positive hermitian slight extension. We will give answers to all these questions.
As an example we will prove that, for each such that, 0 ≤ ≤ 1 , there exists a positive hermitian slight extension ̂ of the Riemann integral R on C([0, 1]), taking the value on the Dirichlet function. Moreover for each there exists ̆ , a maximal positive hermitian slight extension of the Riemann integral, that assumes the value on the Dirichlet function.
We introduce the notion of widely positive extension: roughly speaking a positive hermitian slight extension ̂ is said widely positive if is not possible to extend it to other positive elements of the algebra . Moreover we give the notion of positive regular slight extension that closely reminds the construction of the Lebesgue integral or Segal's construction of noncommutative integration [18]. We will prove that if admits a positively regular absolutely convergent slight extension, which is fully positive, then this extension is unique.
Using the developed ideas we find interesting results concerning Infinite sums and, finally, we show that the Dirac Delta can be studied in the light of the present approach.
Several authors have considered the extension problem for hermitian, positive or representable functionals in various settings and from different points of view [1,2,17,[19][20][21]. Their approach is nevertheless different from the one adopted here.

Preliminary definitions and facts
With the aim of making the paper independent we specialize and repeat to the case of positive hermitian linear functionals the notion of slight extension and give, without proving them, the basic properties.
If is an arbitrary *-algebra, we put Elements of h are called self-adjoint; elements of P( ) are called positive. Clearly, P( ) ⊆ h . We will adopt the following terminology.
In all this paper is a positive hermitian linear functional defined on a dense *-subalgebra 0 of a topological *-algebra [ ] , with continuous involution * .
The functional is said to be closable if one of the two equivalent statements which follow is satisfied. Define • If a → 0 w.r. to and (a ) → , then = 0. • G , the closure of G , does not contain couples (0, ) with ≠ 0.
In this case, the closure is defined on by The closability of implies that is well-defined. The functional is linear and is the minimal closed extension of (i.e., G is closed). Moreover the -continuity of the involution and the hermiticity of on 0 , imply that is hermitian.
Coming back to the general case, let S denote the collection of all subspaces H of × ℂ such that If is nonclosable, i.e. G contains pairs (0, ) with ≠ 0 , then G ∉ S . In this case to every H ∈ S , there corresponds an extension H , to be called a slight extension of , defined on by where, from (g2), is the unique complex number such that (a, ) ∈ H. Moreover, by applying Zorn's lemma to the family S , one proves that admits a maximal slight extension.
Notation: in all this paper we call briefly an extension of any H such that H ∈ S . In the case is closable, G ∈ S , so the corresponding is the unique maximal extension of . Let Given , for every (a, ) ∈ G , there exists a net {a } ⊂ 0 , such that a − →a and (a ) → . The -continuity of the involution and the hermiticity of on 0 , imply that a * − →a * and (a * ) = (a ) → . Hence, (a, ) ∈ G if and only if (a * , ) ∈ G . Then K is a subspace of with the property that a ∈ K if and only if a * ∈ K .
The following proposition holds.

Hermitian and positive extensions
If is nonclosable the extensions defined above are neither hermitian nor positive, in general. It is natural to begin with considering the problem of the existence of hermitian extensions.

Hermiticity
Let H denote the collection of all subspaces H ∈ S for which the following additional condition holds We observe that 0 ⊆ D( H ) ⊆ as vector spaces.

Remark 3.1
As is well-known, an arbitrary element a ∈ can be written in a unique way as a = b + ic , with b and c self-adjoint elements. Since H is a vector space, The following proposition was already given in [3], but the proof was incomplete. We give a new proof here. Proof (i): H satisfies the assumptions of Zorn's lemma. Then it has a maximal element H that defines a maximal hermitian extension ̆.
(ii): The result is obvious if is closable. Let now be nonclosable. As it is clear, for every hermitian extension ̆ , one has D(̆) ⊆ K . Let, by contradiction, a ∈ K ⧵D(̆) . Then also a * ∈ K ⧵D(̆) since a ∈ D( H ) ⇔ a * ∈ D( H ) .
, then at least one, between b or c, does not belong to D( H ) . Let, without loss of generality, b ∈ K ⧵D(̆) . Since b ∈ K , by Lemma 2.5, we can choose ∈ ℝ such that (b, ) ∈ G . Consider Ğ⊕ ⟨(b, )⟩ , where ⟨(b, )⟩ denotes the subspace generated by (b, ) . We will prove that Ğ⊕ ⟨(b, )⟩ ∈ H and this contradicts the maximality of ̆ . Thus we need to show that if (a 1 , 1 If 0 is a proper subspace of K then, as we have seen in the previous proof of (ii), there exists b = b * ∈ K ⧵ 0 , and moreover, ∀ ∈ ℝ ,

Remark 3.5
If we impose a constraint to an extension and then we are looking for a maximal element, we will find, in general, a smaller domain. In our case the situation is very different: as we have shown a maximal hermitian extensions of is a maximal extension.  we say that ̂ is fully positive if ̂ is positive and D(̂ ) ∩ P( ) = K + .

Remark 3.7
Since the domain of a maximal extension ̆ is K , then we deduce that if a maximal extension ̆ is positive, then ̆ is fully positive.
Definition 3.8 Given , we define P as the collection of all subspaces K ∈ H satisfying the following additional condition (p3) (a, ) ∈ K and a ∈ P( ) , implies ≥ 0.
Since is positive, then P ≠ ∅ and G ⊆ K ⊆ G for every K ∈ P .
To every K ∈ P , there corresponds a hermitian extension K of , defined on by where, from (g2), of Sect. 2, is the unique complex number such that (a, ) ∈ K . By (p3), K is a positive hermitian extension of . We observe that 0 ⊆ D( K ) ⊆ as vector spaces.
Since P satisfies the assumptions of Zorn's lemma, we have the following With this in mind, if ̂ is a positive hermitian extension of and c ∈ h , we introduce the following notations that will use to characterize both the elements for which it is possible to find a positive hermitian extension and, given such an element, the values this extension may assume.
where we put c,̂ ∶= +∞ if the set in the right hand side of the definition is the empty set; From Remark 3.12, we deduce the following 14 Let ̂ be a positive hermitian extension of and let c ∈ K + . Then An immediate consequence of Proposition 3.13 is the next lemma.
◻ Corollary 3.20 Let ̂ be a positive hermitian extension of . If ̂ is fully positive then K ‡ = K + and ̂ is widely positive. In particular if is fully positive then K ‡ = K + .
Proof If ̂ is fully positive then (by Lemma 3.19), we have ◻ An important result which follows from the previous discussion, shows that the sole elements c ∈ K + ⧵ 0 for which we can find a positive hermitian extension of , are exactly those with finite c, . More exactly the following theorems hold.

Theorem 3.22 Let be nonclosable, let ̂ be a positive hermitian extension of and let
Proof We first show that, ∀a ∈ D(̂ ), ∈ ℝ , such that, a + c ∈ P( ) , we have ̂ (a) + ≥ 0 . The case = 0 is trivial, so we can distinguish two cases: < 0 and > 0.
The next theorem establishes that if is nonclosable, then any maximal positive hermitian extension ̆ of , has "all possible" (see Lemma 3.19) positive elements in its domain.

Remark 3.29
The previous theorem shows that, for each such that, 0 ≤ ≤ 1 , there exists a positive hermitian extension ̂ of the Riemann integral R on C ([0, 1]), taking the value on the Dirichlet function c, despite of the Lebesgue integral of c being equal to 0. Obviously, ̂ is neither an extension of the Lebesgue integral, nor depends on it.

Absolutely convergent extensions
Now we will require the *-algebra to satisfy further conditions.

Definition 3.30
Let be a *-algebra. We say that has the property (D) if, for every a ∈ h , there exists a unique pair (a + , a − ) of elements of , with a + , a − ∈ P( ) such that

Then we put
In what follows we suppose that has the property (D). In this case, one has:  |a| ∶= a + + a − .
Then we are induced to give the following Definition 3.35 Let be absolutely convergent. We define AC the subfamily of P , whose elements K satisfy the following additional condition: (p ac ) (a, ) ∈ K , with a ∈ h , implies that ∃ 1 , 2 ≥ 0 such that, (a + , 1 ), (a − , 2 ) ∈ K.
Since K is a vector space verifying condition (g2) of Sect. 2, then = 1 − 2 , and since is absolutely convergent, then AC ≠ ∅ and G ⊆ K ⊆ G , for every K ∈ AC . To every K ∈ AC , there corresponds a hermitian extension K of , defined on by where, from (g2), of Sect. 2, is the unique complex number such that (a, ) ∈ K . By (p ac ), K is an absolutely convergent extension of .
We observe that 0 ⊆ D( K ) ⊆ as vector spaces.

Theorem 3.36 If is absolutely convergent, then has a maximal absolutely convergent extension.
Proof The family AC satisfies the assumptions of Zorn's lemma, hence it has a maximal element to which there corresponds a maximal absolutely convergent extension. ◻

Proposition 3.37 Let ̆ be a maximal positive hermitian extension of . If ̆ is absolutely convergent, then D(̆) is span {K ‡ }.
Proof If ̆ is a maximal positive hermitian extension, then, by Theorem 3.24, ̆ is widely positive, so the statement follows by the previous Corollary 3.34. ◻ Now we state the following important theorem and corollary.

Positively regular extensions
In this section we will construct a particular extension ̂ of a hermitian positive linear functional , following essentially the model of the construction of the Lebesgue integral. We recall that for a, b ∈ h , we have defined Then we have: Indeed |a| − 0 ∈ P( ), |a| − a = 2a − ∈ P( ) and |a| + a = 2a + ∈ P( ).

Definition 3.40 An extension ̂ of is said to be positively regular if
We observe that, by definition, a positively regular extension is positive.
To obtain a positively regular extension ̂ of , we start from the next Definition 3.41 Let be absolutely convergent and positively regular. For a ∈ , let a = a 1 − a 2 + i(a 3 − a 4 ), the unique writing of a, with a i ∈ P( ), 1 ≤ i ≤ 4 . Then we define PR as the subfamily of AC , whose elements K satisfy the following additional condition: We observe that, as before, since is absolutely convergent and positively regular, then PR ≠ ∅ and, for every K ∈ PR , G ⊆ K ⊆ G . Moreover, since K is a vector space verifying condition (g2) of Sect. 2, then = 1 − 2 + i( 3 − 4 ).
To every K ∈ PR there corresponds an extension K of , defined on by where, from (g2) of Sect. 2, is the unique complex number such that (a, ) ∈ K . By definition, K is a positively regular absolutely convergent extension of . Again we observe that 0 ⊆ D( K ) ⊆ as vector spaces. Invoking Zorn's Lemma we have the following Theorem 3.42 If is absolutely convergent and positively regular, then has a maximal positively regular absolutely convergent extension.

D( K ) = {a ∈ ∶ (a, ) ∈ K}
Now we state the following

Three simple examples
We give now some easy examples, without going into the details of the proofs.

Example: the Lebesgue integral
The Henstock-Kurzweil integral is an extension of the Lebesgue integral and it is possible to verify it applying the abstract method developed in this section. We use all the conventions given in the introduction and let, ∶= R , be the Riemann integral on I. Then the Lebesgue integral on I is a positive hermitian extension of .
In this case, there exist many possible extensions of the Lebesgue integral. We consider in what follows the Henstock-Kurzweil (HK) integral. The fact that the HK integral includes the Lebesgue integral was proved by Henstock [7,8].
In [3] it was proved that the HK integral is not a maximal positive hermitian extension of the Riemann integral. From Theorem 3.9 we have the following

Theorem 4.1 There exists a maximal positive hermitian extension of the Henstock-Kurzweil integral.
Moreover, the approach proposed here allows us to give a theoretical proof of the existence of a maximal hermitian positive slight extension for the Henstock-Kurzweil integral opening the challenge of finding it explicitly.

Example: infinite sums
Let denote the complex vector space of all infinite sequences of complex numbers. is a *-algebra if the product ⋅ of two sequences = (a k ) , = (b k ) , k ≥ 1 , is defined component-wise and the involution by * = (a k ) . Let us endow with the topology defined by the set of seminorms Let 0 denote the *-subalgebra of consisting of all finite sequences in the sense that = (a k ) ∈ 0 if, and only if, there exists N ∈ ℕ such that a k = 0 if k > N . We define The symbol of series is only graphic since all sums are finite.
This functional, which is obviously positive hermitian, is nonclosable. To see this let us consider the sequence of sequences ( n ) = ((a n,k )) ⊆ 0 with, for n ≥ 1 , a n,k ∶= n,k (the Kronecker delta).
For fixed k, clearly lim n→∞ a n,k = 0 . Hence n → as n → ∞ and, applying , we get We observe that any convergent series which converges to l ∈ , can be "rewritten" as a sequence of sequences ( n ) ⊆ 0 , with n → and ( n ) → l , as n → ∞.
The next proposition shows that in this case K is not a proper subset of the algebra.

Proposition 4.2
Let and be as above. Then 0 is a dense subalgebra of with K = .

Proposition 4.3 The functional admits a maximal hermitian extension ̆ which is a maximal extension with D(̆) = . ◻
As discussed above, there exists infinitely many extensions of , the procedure of taking the limit of the partial sums s n ∶= a 1 + a 2 + ⋯ + a n , being just one of them. This is historically a very well known fact which dates back to the Grandi series Grandi asserted that this infinite sum is equal to 1 2 . Now we know that this can be obtained via Ramanujan sums. But also, more elementary, considering the following extensions of .

Hölder summation
The first (historically rigorous) extension of , due to A.L. Cauchy, is the following. Given a sequence = (a k ) ∈ 0 , define by induction: s n ∶= a 1 + a 2 + ⋯ + a n , ∀n ≥ 1. We observe that if we impose, in defining D( H ) , h = 0 , we obtain 1 . If we impose h ≤ 1 we obtain the so called Cesàro summation. Moreover a direct calculation shows that the Cesàro sum of Grandi's series is defined and its value is whereas (1, −1, 1, −1, …) ∉ D( 1 ).

Abel summation
Given a sequence = (a k ) , we define an extension A of with domain defined by Clearly A ( ) = a 1 + a 2 + ⋯ + a N , if ( ) ∈ 0 and it easy to see that A is a positive hermitian extension of . Now take = ((−1) k+1 ) . Then we obtain H 0 n =a 1 + a 2 + ⋯ + a n , Taking the limit for x → 0 + , we get Indeed it is well known that the Abel summation is a generalization of the Hölder summation or still better, in light of our approach, the Abel summation is a positive hermitian extension of the Hölder summation.

Positive hermitian extensions
As we have seen, Hölder summation and Abel summation are both positive hermitian extensions of and, moreover they are extension of 1 . At this point the following question arises: is the classical definition of the sum of a series really natural?
The answer is yes, if we are looking to positive hermitian extensions of . We start with the following corollary which is a direct consequence of Proposition 4.2.

Corollary 4.5
Given the algebra , we have K + = P( ) . ◻ With the next proposition given ∈ K + ⧵D( ) , we find explicitly , and , .
Proof From Corollary 4.5 and Proposition 4.6 we have Now it seems interesting to us to show another example in which K coincides with the entire algebra . Starting with a subalgebra of and changing the topology with a finer one, we will find a new topological *-algebra 1 . Then, taking the closure of 0 in 1 , we will obtain the required algebra ⊆ 1 .
Let us consider the subalgebra 1 ⊆ of all bounded sequences, endowed with the norm Then 1 is a topological *-algebra with 0 ⊆ 1 . Now we find the closure of 0 in 1 .
With the next proposition we will find K .

Proposition 4.10
Let , 0 and be as above. Then K = .

Example: the Dirac delta
Let us consider the Banach convolution algebra L 1 (ℝ) with its usual norm, involution and multiplication; i.e, for f , g ∈ L 1 (ℝ) Let C c (ℝ) denote the *-algebra of continuous functions with compact support. Then, C c (ℝ) is a *-subalgebra of L 1 (ℝ).
While the left hand side could be meaningless for f ∈ L 1 (ℝ) , the right hand side could produce a finite number for certain f ∈ L 1 (ℝ).
Let us put and define First, we observe that D(̂ ) ⊊ L 1 (ℝ) . Indeed, the function obviously belongs to L 1 (ℝ) ; but So we have the following

Remark 4.15
The linear functional ̂ is a positive hermitian extension of .
Acknowledgements The authors thank the referee for his/her useful comments and suggestions. This work has been done in the framework of the activity of GNAMPA-INDAM.
Funding Open access funding provided by Università degli Studi di Palermo within the CRUI-CARE Agreement.
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