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This is primarily a request for references and advices.

Question (edited on 10/29/2011). What's known about comprehensive generalisations of Gelfand's spectral theory for unital [associative] normed algebras [over the real or complex field] (*)?

Here, a generalisation should be meant as a framework, say, with the following distinctive features (among the others):

  1. It should be founded on somehow different bases than the classical theory - especially to the extent that the notion itself of spectrum isn't any longer defined in terms of, and cannot be reduced to, the existence of any inverse in some unital algebra.
  2. It should recover (at least basic) notions and results from the classical theory for unital Banach algebras in some appropriate incarnation (more details on this point are given below), for which the "generalised spectrum" does reproduce the classical one.
  3. It should be unsensitive to completeness [under suitable mild hypotheses] in any setting where completeness is a well-defined notion (**), so yielding as a particular outcome that an element in a unital normed algebra, $\mathfrak{A}$, shares the same spectrum as its image in the Banach completion of $\mathfrak{A}$.

(*) If useful to know, my absolute reference here is the (let me say) wonderful book by Charles E. Rickart: General Theory of Banach Algebras (Academic Press, 1970).

(**) At least in principle, the kind of generalisation that I've in mind is tailored on the properties of topological vector spaces, though I've worked it out only in the restricted case of normed spaces.


Historical background.

As acknowledged by Jean Dieudonné in his History of Functional Analysis, the notion of spectrum (along with the foundation of modern spectral theory) was first introduced by David Hilbert in a series of articles inspired by Fredholm's celebrated work on integral equations (the word spectrum seems to have been lent by Hilbert from an 1897 article by Wilhelm Wirtinger) in the effort of lifting properties and notions from matrix theory to the broader (and more abstract) framework of linear operators. Especially, this led Hilbert to the discovery of complete inner product spaces (what we call, today, Hilbert spaces just in his honour). In 1906, Hilbert himself extended his previous analysis and discovered the continuous spectrum (already present but not fully recognised in earlier work by George William Hill in connection with his own study of periodic Sturm-Liouville equations).

A few years later, Frigyes Riesz introduced the concept of an algebra of operators in a series of articles culminating in a 1913 book, where Riesz studied, among the other things, the algebra of bounded operators on the separable Hilbert space. In 1916 Riesz himself created the theory of what we call nowadays compact operators. Riesz's spectral theorem was the basis for the definitive discovery of the spectral theorem of self-adjoint (and more generally normal) operators, which was simultaneously accomplished by Marshall Stone and John von Neumann in 1929-1932.

The year 1932 is another important date in this story, as it saw the publication of the very first monography on operator theory, by Stefan Banach. The systematic work of Banach gave new impulse to the development of the field and almost surely influenced the later work of von Neumann on the theory of operator algebras (developed, partly with Francis Joseph Murray, in a series of articles starting from 1935). Then came the seminal work of Israil Gelfand (partly in collaboration with Georgi E. Shilov and Mark Naimark), who introduced Banach algebras (under the naming of normed rings) and elaborated the corresponding notion of spectrum starting with a 1941 article in Matematicheskii Sbornik.

Now, it is undoubtable that Gelfand's work has deeply influenced the subsequent developments of spectral theory (and, accordingly, functional analysis). Yet, as far as I can understand in my own small way, something is still missing. I mean, something which may still be done, on the one hand, to clean up some inherent "defects" (or better fragilities) of the classical theory and, on the other, to make it more abstract and, then, portable to different contexts.

Naïve stuff.

As I learned from an anonymous user on MO (here), the term spectrum, in operator theory as well as in the context of normed algebras, is seemingly derived from the Latin verb spècere ("to see"), from which the root spec- of the Latin word spectrum ("something that appears, that manifests itself, vision"). Furthermore, the suffix -trum in spec-trum may come from the Latin verb instruo (like in the English word "instrument", which follows in turn the Latin noun instrumentum). So, the classical (or, herein, Gelfand) spectrum may be really considered, even from an etymological perspective, as a tool to inspect (or get improved knowledge of) some properties. I like to think of it as a sort of magnifying glass; we can move it through the algebra, zoom in and out on its elements, and get local information about them and/or global information about the whole structure.

Now, taking in mind (some parts of) another thread on this board about "wrong" definitions in mathematics, we are likely to agree that the worth of a notion is also measured by its sharpness (let me be vague on this point for the moment). And the classical notion of spectrum is, in fact, so successful because it is sharp in an appropriate sense, to the extent that it reveals deep underlying connections, say, between the algebraic and topological structures of a complicated object such as a Banach algebra (which is definitely magic, at least in my view). On another hand, what struck my curiosity is the consideration that the same conclusion doesn't hold (not at least with the same consistency) if Banach algebras are replaced by arbitrary (i.e. possibly incomplete) normed algebras, where the spectrum of a given element, $\mathfrak{a}$, can be scattered through the whole complex plane (in the complete case, as it is well-known, the spectrum is bounded by the norm of $a$, and indeed compact). So the question is: Why does this happen? And my answer is: essentially because the classical notion of spectrum is too algebraic, though completeness can actually conceal its true nature and make us even forgetful of it, or at least convinced that it must not be really so algebraic (despite of its own definition!) if it can dialogue so well with the topological structure. Yes, any normed algebra can be isometrically embedded (as a dense subalgebra) into a Banach one, but I don't think this makes a difference in what I'm trying to say, and it does not seriously explain anything. Clearly enough, the problem stems from the general failure in the convergence of the Neumann series $\sum_{n=0}^\infty (k^{-1}\mathfrak{a})^n$ for $k$ an arbitrary scalar with modulus greater than the norm of $\mathfrak{a}$. And why this? Because the convergence of such a Neumann series follows from the cauchyness of its partial sums, which is not a sufficient condition to convergence as far as the algebra is incomplete. According to my humble opinion, this is something like a "bug" in the classical vision, but above all an opportunity for getting a better understanding of some facts.

Motivations.

In the end, my motivation for this long post is that I've seemingly developed (the basics of) something resembling a spectral theory for linear (possibly unbounded) operators between different normed spaces. To me, this stuff looks like a sharpening of the classical theory in that it removes some of its "defects" (including the one addressed above); and also as an abstraction since, on the one hand, it puts standard notions from the operator setting (such as the ones of eigenvalue, continuous spectrum, and approximate spectrum) on a somehow different ground (so possibly foreshadowing further generalisations) and, on the other, it recovers familiar results (such as the closeness, the boundedness, and the compactness of the spectrum as well as the fact that all the points in the boundary are approximate eigenvalues) as a special case (while revealing some (unexpected?) dependencies).

Then, I'd really like to know what has been already done in these respects before putting everything in an appropriate form, submitting the results to any reasonable journal, and being answered, possibly after several months, that I've just reinvented the wheel. It would be really frustrating... Yes, of course, I've already asked here around (in Paris), but I've got nothing more concrete than contrasting (i.e. negative and positive) feelings. Also, I was suggested to contact a few people, and I've done it with one of them some weeks ago (sending him something like a ten page summary after checking his availability by an earlier email), but I've got no reply so far and indeed he seems to have disappeared... Then, I resolved to come here and consult the "oracle of MO" (as I enjoy calling this astounding place). :-)

Thank you in advance for any help.

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Small point: I don't think there's anything unnatural about the idea that if A is a dense subalgebra of B (neither necessarily complete) then the $A$-spectrum of an element $a\in A$ can differ from the $B$-spectrum of $a$ in $B$. That's just a "fact of life" –  Yemon Choi Oct 28 '11 at 22:04
    
Could you perhaps edit your "post" to highlight what your specific question or request is? There seems to be a lot of background justification, and not much in the way of an actual "question" –  Yemon Choi Oct 28 '11 at 22:06
    
I'm also somewhat unconvinced by this passage: "So the question is: Why does this happen? And my answer is: essentially because the classical notion of spectrum is too algebraic, though completeness can actually conceal its true nature and make us even forgetful of it, or at least convinced that it must not be really so algebraic (despite of its own definition!) if it can dialogue so well with the topological structure" –  Yemon Choi Oct 28 '11 at 22:07
    
@Yemon Choi. Thanks for your comments. I see your point, and I've nothing to object. Yet, my view is different, or - better - complementary. I'm mainly interested in approximation theory, where the (still naïve) basic idea is, say, that you're not really interested in solving your equations in an "exact way", but you content to do it within the bounds of an arbitrary $\varepsilon > 0$. So, in this setting (as well as in others), you may want to rule out any unnecessary complications arising from the fact that favorable properties are simply lacking (as a "fact of life"), and hence make (...) –  Salvo Tringali Oct 29 '11 at 10:28
    
(...) your framework somehow more flexible (starting with your definitions). That's that. Also, I should have probably remarked that my reflection about completeness is not the true point here, though it served as a trigger (and it is indeed a feature of the kind of generalisation that I've in mind). One thing for sure, I'm going to edit the OP to second your suggestion, highlight my requests, and (try to) clean up some passages. As for the rest, could you elaborate your last comment above? Thanks again. –  Salvo Tringali Oct 29 '11 at 10:40
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1 Answer 1

There exists an approach by Berezansky and Brasche to define a kind of generalized selfadjointness and develop some eigenfunction expansions for operators between different Hilbert spaces. This is done within the theory of rigged Hilbert spaces. Their motivation was apparently different from yours - they wanted to cover the case of Schroedinger operators with distribution potentials. Nevertheless there can be some connection. See

Yu.M. Berezansky and J. Brasche, Generalized selfadjoint operators and their singular perturbations. Methods Funct. Anal. Topol. 8, No. 4, 1-14 (2002);

Yu.M. Berezansky, J. Brasche, and L. P. Nizhnik, On generalized selfadjoint operators on scales of Hilbert spaces, Methods Funct. Anal. Topol. 17, No. 3, 193-196 (2011).

The second paper is available online: http://www.imath.kiev.ua/~mfat/

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@Anatoly Kochubei. Thanks for your answer and the references. Indeed, the 2nd paper is interesting to me, though it's still, say, too "localised" with respect to the kind of stuff that I've in mind. In any case, I will read it with more attention. Also, let me remark once more that I've proved so far only basic results (essentially the ones listed in the OP and a few minor others). For one thing, I did not even have a way to show that, under reasonable assumptions, this "generalized spectrum" is non-empty. –  Salvo Tringali Oct 29 '11 at 15:27
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