On complementing topological algebras

This paper is based on a part of my thesis written under the supervision of Professor A. Mallios (University of Athens). I wish to express my sincere thanks to him for constant help and stimulating advice during the preparation of this work.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 49, Functional Analysis-4, 1997.     

On ideals of free lie algebras

The only possible ideals which are finitely generated as subalgebras of a free Lie algebra are the zero ideal or the whole algebra. This should not be understood as “all nontrivial ideals which have infinite rank are proper ideals”.In this paper we prove that a nontrivial ideal which has infinite rank may be the whole algebra under certain conditions.     

On inner ideals in ternary algebras

Research supported by a grant from the United Kingdom Science and Engineering Research Council     

On primary ideals in group algebras

LetG be a -compact locally compact group such thatG/Z(G), whereZ(G) denotes the center ofG, has a relatively compact commutator subgroup. It is shown that primary ideals inL ¹(G) are maximal.     

On Σ-nilpotent ideals of topological PI-rings

We show that under certain conditions on the topology of a faithful module M over a topological PI-ring R, if M has at most countable dual topological Krull dimension, then the closure of the sum of all Σ-nilpotent ideals of the ring R is a Σ-nilpotent ideal too, and in the case of a bounded ring R its topological Baer radical is Σ-nilpotent. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 111–118, 2006.     

On identifying the maximal ideals in Banach algebras

We consider the problem of identifying the maximal ideals of a Banach algebra S that is contained in a larger Banach algebra B. If S is dense in B in an appropriate sense and if the spectral radii in S and B are the same, then S and B have the same maximal ideals. The result is illustrated by two examples of Banach algebras S in which the density and spectral radius conditions are easily shown to be valid with respect to a larger algebra B whose maximal ideals are known. The first example is a convolution algebra on a group where the functions in the algebra have specified rate of decay. The second example is a generalized version of an algebra introduced by I. Hirschman. Two applications of the generalized Hirschman algebra are presented: these relate to filtering and prediction of stationary random sequences and concern the asymptotic behavior of the errors incurred by the finite memory predictor and by the finite lag filter.     

Closed ideals in topological algebras: A characterization of the topological Φ-algebra C k (x)

Let A be a uniformly closed and locally m-convex Φ-algebra. We obtain internal conditions on A stated in terms of its closed ideals for A to be isomorphic and homeomorphic to C _k (X), the Φ-algebra of all the real continuous functions on a normal topological space X endowed with the compact convergence topology.     

On the Bishop boundary in topological algebras

We characterize the existence of the Bishop boundary in the context of an Urysohn (topological) algebra, by means ofG _δ points. Considering a subset of its spectrum and the restriction of its Gel'fand transform algebra on the previous subset, we realize its Bishop boundary, as the trace of the Bishop boundary of the initial algebra on the spectrum of the restriction algebra. We get a generalization of the latter by considering a continuous algebra morphism. This paper is based on and extends results of the author's Doctoral Thesis, written under the supervision of Professor A. Mallios (Univ. of Athens).     

On compact multipliers of topological algebras

It is shown that if the maximal ideal space (A) of a semisimple commutative complete metrizable locally convex algebra contains no isolated points, then every compact multiplies is trivial. In particular, compact multipliers on semisimple commutative Fréchet algebras whose maximal ideal space has no isolated points are identically zero.     

On the structure of inner ideals of nest algebras

IfA is a nest algebra andA _s=A ∩ A* , whereA* is the set of the adjoints of the operators lying inA, then the pair (A, A _s) forms a partial Jordan *-triple. Important tools when investigating the structure of a partial Jordan *-triple are its tripotents. In particular, given an orthogonal family of tripotents of the partial Jordan *-triple (A, A _s), the nest algebraA splits into a direct sum of subspaces known as the Peirce decomposition relative to that family. In this paper, the Peirce decomposition relative to an orthogonal family of minimal tripotents is used to investigate the structure of the inner ideals of (A, A _s), whereA is a nest algebra associated with an atomic nest. A property enjoyed by inner ideals of the partial Jordan *-triple (A, A _s) is presented as the main theorem. This result is then applied in the final part of the paper to provide examples of inner ideals. A characterization of the minimal tripotents as a certain class of rank one operators is also obtained as a means to deduce the principal theorem.     

On the sheaf theory

The“Gel’fand sheaf” of a topological algebra is endowed with auniform structure, this being complete if and only if, the spectrum of the algebra considered is complete. Examples are also provided.