Class Of Topological Algebras Research Articles

On TQ-Algebras

A topological algebra A is called a Q-algebra if its set of quasi-invertible elements (of invertible elements, if A is unital) is an open set. This class of topological algebras has been extensively studied by several authors, but mostly in unital or commutative case. Properties of (not necessarily unital or commutative) TQ-algebras and, in the special case, of TQ-algebras with functional topological spectrum are given in the present paper.

Generalization of some properties of Banach algebras to fundamental locally multiplicative topological algebras

In this article we generalized some properties of Banach algebras, to a new class of topological algebras namely fundamental and fundamental locally multiplicative topological algebras (abbreviated by FLM). Also the new notion of sub-multiplicatively metrizable topological algebra is given and some well known spectral properties of Banach algebras are generalized to such kind of algebras.

The exponential map in fundamental locally multiplicative topological algebras

A class of topological algebras, which we call it a fundamental locally multiplicative one, has already been introduced and the exponential map has been defined before on complete metrizable fundamental locally multiplicative topological algebras. Here, in this note we prove the main identification about this map. Mathematics Subject Classification: Primary 46H; Secondary 46A

Small deformations of topological algebras

We investigate stability of various classes of topological algebras and individual algebras under small deformation of multiplication.

Semisimple classes and lower radicals of topological nonassociative algebras

In this article semisimple classes of topological algebras are characterized in a series of varieties of rings (in particular, all subvarieties of the varieties of alternative and Jordan algebras). Characterizations of semisimple classes of hereditary radicals are obtained, and the heredity of semisimple classes of radicals is demonstrated. It is proved that the construction of lower radicals stabilizes on the first infinite ordinal in the varieties of topological algebras considered.

Multiplicative functionals and a class of topological algebras

Every multiplicative linear functional on a pseudocomplete locally convex algebra satisfying the “sequential” property of Husain and Ng is bounded (a topological algebra is called “sequential” if every null sequence contains an element whose powers converge to zero). Characterizations of such algebras are given, with some examples.

On the generalized spectra of topological algebras

Local equicontinuity of the generalized spectra of certain topological algebras implies that the spectra in question coincide as topological spaces with the generalized spectra of the respective completed topological algebras [7]. On the other hand, for a fairly large class of topological algebras (i.e., m-barreled locally convex algebras), local equicontinuity of the (usual) spectrum is equivalent to its local compactness [6]. In this paper we are concerned with generalized spectra of (nonnecessarily commutative [II]) topological algebras. The results obtained have a special bearing, for the case under consideration, on the corresponding results contained in [6, 7, 1 I], which were also the motivation to the present setting. Thus, let E and F be topological algebras and let A(E, F) be the generalized spectrum of E (for F given) (i.e., the set of nonsero continuous algebra homomorphisms of E into F topologized as a subset of P8(E, F); cf. Section 3 below). It is shown that if E is an m-barreled locally convex algebra and F a locally m-convex semiMonte1 one, local equicontinuity of d(E, F) is equivalent to its local compactness (cf., Theorem 3.2). Furthermore, we are dealing with representations of the generalized spectra of topological tensor product algebras. In this respect, it is proved that there exists a bicontinuous injection of the generalized spectrum of a locally convex infinite tensor product algebra into the Cartesian product space of the spectra of the factor algebras (cf., Theorem 5.2). Moreover, by considering completed topological algebras, we still assume local equicontinuity for the generalized spectra. This assumption yields analogous representations of the generalized spectra of certain completed locally convex infinite tensor product algebras (with continuous multiplication) (Theorem 5.3). On the other hand, with the class of m-barreled locally convex algebras several aspects concerning the continuity of the Gel’fand map are closely related [6]. Thus, we are also interested therein in the continuity of the generalized Gel’fand map (cf. Section 4). The results obtained constitute extended forms of those of [6; Section 31.

On a class of topological algebras

On a class of topological algebras

On the spectra of topological algebras

Equicontinuity of the spectra of topological algebras was proved to be of a particular significance in connection with topological tensor product algebras, especially when (topological) completeness of such algebras had to be considered (cf., for instance, [II]. [I7]). On the other hand, it was also proved [13] that only local equicontinuity [3] is needed, whenever, in this respect, equicontinuity has been used instead. We consider in this paper a fairly large class of topological algebras, i.e., m-barreled locally convex algebras (cf. Section 2), for which local equicontinuity of the spectrum is equivalent to its local compactness. FrCchet algebras with locally compact spectrum, e.g., Stein algebras (cf. Section 4 below), and, more generally, topological inductive limits of such algebras belong to this category of algebras. The situation described comprises a recent result in ([3]; p. 108, Theorem 3.1), which was also the motivation to the present setting. We consider, furthermore, an application of the preceding concerning the spectrumof certain infinite tensor product topological algebras [12]. An other application of the said result in [3] combined with the theory of topological tensor product algebras [II] is given by considering topological tensor products of Stein algebras (cf. Section 4 below). Finally, a particular class of the topological algebras dealt with in the following has also been recently considered in [IL?]. Thus, Section 3 below gives information concerning the continuity of the Gel’fand map for the topological algebras involved, which extend and improve the corresponding results in [ZS].