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):

• 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.
• 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.
• 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.

In the end, my motivation for this long post is that I've seemingly developed (the basics of) something like resembling a spectral theory for linear (possibly unbounded) operators between different normed spaces, indeed linear (possibly discontinuous) operators between different topological vector 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).

3 More minor corrections

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 $\sum_{n=0}^\infty (k^{-1}a)^n$ k^{-1}\mathfrak{a})^n$for$k$an arbitrary scalar with modulus greater than the norm of$a$. \mathfrak{a}$. 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.

In the end, my motivation for this long post is that I've seemingly developed (the basics of) something like a spectral theory for linear (possibly unbounded) operators between different normed spaces, indeed linear (possibly discontinuous) operators between different topological vector 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 possible 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).

2 Minor corrections

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 missingin this (wonderful) picture. I mean, something which may still be done, on the one hand, to clean up the notion itself of spectrum (as given in the classical framework of normed algebras) of 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.

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