The main result of this paper is a characterization of the Strongly algebraically closed algebras in the  lattice of all real-valued continuous functions and the equivalence classes of $\lambda$-measurable. We shall provide conditions  which Strongly algebraically closed algebras carry a strictly positive Maharam submeasure. Particularly, it is proved that if $B$ is a Strongly algebraically closed lattice  and $(B,\, \sigma)$ is a Hausdorff space  and $B$ satisfies  the   $G_\sigma$ property, then $B$ carries a strictly positive Maharam submeasure.

Let A be a finite dimensional k-algebra and R be a locally bounded category such that R®R/G=A is a Galois covering defined by the action of a torsion-free group of automorphisms of R. Following [30], we provide criteria on the convex subcategories of a Strongly simply connected category R in order to be a cycle-finite category and describe the module category of A. We provide criteria for A to be of polynomial growth.

Let G be a locally compact abelian group and µ ε M (G) be given. If µ is the product of an invertible and an idempotent measure then the ideal µ * L1 (G) is obviously closed in L1 (G). The problem whether the converse is also true, which was raised by E. Hewitt, has been first studied by I. Glicksberg and later by B. Host and F. Parreau who completely solved it in their impressive paper. Thus µ * L1 (G) is closed in L1 (G) if and only if µ is the product of an invertible and an idempotent measure in M (G). In this paper we study the multipliers with closed range on the weighted group algebra L1 (G, ω). We prove among other things that, the Glicksberg-Host-Parreau theorem for group algebras remain valid for regular weighted group algebras.    

Let S be a foundation semigroup with identity and Ma(S) be its semigroup algebra. In this paper, we give necessary and su cient conditions for the existence of a bounded approxi-mate identity in closed codimension one ideals of semigroup algebra Ma(S) of a locally compact topological foundation semigroup with identity.

The paper continues the study of the authors on relationships between topological systems of S. Vickers and attachments of C. Guido. We extend topological systems to algebraically-topological systems. A particular instance of the latter, called attachment system, incorporates the notion of attachment, thus, making it categorically redundant in mathematics. We show that attachment systems are equipped with an internal topology, which is similar to the topology induced by locales. In particular, we provide an attachment system analogue of the well-known categorical equivalence between sober topological spaces and spatial locales.

Let S be a locally compact foundation semigroup with identity and ( ) be its semigroup algebra. Let X be a weak*-closed left translation invariant subspace of ( ) In this paper, we prove that X is invariantly complemented in ( ) if and only if the left ideal of ( ) has a bounded approximate identity. We also prove that a foundation semigroup with identity S is left amenable if and only if every complemented weak*-closed left translation invariant subspace of ( ( )) is invariantly complemented in ( ( )).

In this paper we show that a bounded linear operator on a Banach space X is polaroid if and only if p(T) is polaroid for some polynomial p. Consequently, algebraically paranormal operators defined on Banach spaces are hereditarily polaroid. Weyl type theorems are also established for perturbations ¦(T+K), where T is algebraically paranormal, K is algebraic and commutes with T, and f is an analytic function, defined on an open neighborhood of the spectrum of T+K, such that f is nonconstant on each of the components of its domain. These results subsume recent results in this area.