Spectral geometry of non-local topological algebras

The classical problem of existence of non-local function algebras was settled in the affirmative by Eva Kallin in the early sixties by her well-known example [17], (see also [6, p. 170] and [22, p. 83, Example]. A few years later R. G. Blumenthal [3, 4] remarked that Kallin’s example was simply a particular case of a type of algebras studied by S. J. Sidney in his dissertation (see [21]). The previous results were obtained within the standard context of Banach function algebra theory. On the other hand, working within the general framework of Topological Algebras, not necessarily normed ones (we refer to A. Mallios [18] for the relevant terminology), we have already considered in [12] the spectrum of Sidney’s algebra. More precisely, we looked at it, as a ”gluing space” of the spectra of two factor tensor product algebras, whose sum constituted, by definition, the algebra of Sidney. In point of fact, it was Blumenthal (loc. cit.), who actually defined the spectrum of Sidney’s algebra, as a gluing space, his result being thus subsumed into ours [12, Theorem 5.2]. Now, continuing herewith our previous work in [12], we further obtain a general existence theorem for non-local topological algebras (a la Blumenthal; see Theorem 3.2). Furthermore, based on a recent article of R. D. Mehta [19], still within the Banach function algebra theory, we consider the Choquet boundary of the (generalized) algebra of Sidney (cf. Theorem 4.1 in the sequel). Indeed, by changing the hypotheses, appropriately, we are able to have the same boundary in a more concrete form, than that one in [12]. Yet, following in the preceding general set-up A. Mallios [18] (see Lemma 4.1 below), we also obtain the Silov boundary of the same Sidney’s algebra, as above.