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We describe all those commutative Fr echet algebras which may be contin- uously embedded in the algebra C((X)) in such a way that they contain the polynomials. It is shown that these algebras (except C((X)) itself) always satisfy a certain equicontinu- ity condition due to Loy. Using this result, some applications to the theory of automatic continuity are given; in particular, the uniqueness of the Fr echet algebra topology for such algebras is established.

Introduced in the early 80’s by Happel-Ringel [4], the class of tilted algebras has played an important role in the development of the representation theory of algebras. However, it is not always easy to identify a tilted algebra looking, for instance, at its ordinary quiver. On the other hand, the tilted algebras behave nicely with respect to their semi-convex subcategories, that is, if A is a tilted algebra and B is a semi-convex subcategory of A, then also B is tilted. Recall that, given an algebra A, a subcategory B of A is semi-convex provided there is a sequence B = Bs, · · · , B0 = A such that Bi = Ci[M ′ i ] (respectively, Bi = [M ′ i ]Ci) is a one-point (co-)extension of a convex subcategory Ci of Bi−1 = Ci[M ′ i ⊕ M ′′ i ] (respectively Bi−1 = [M ′ i ⊕M ′′ i ]Ci) by a Ci-module M ′ i , possibly M ′′ i = 0. Clearly, convex subcategories are semi-convex. By looking at the minimal non-tilted algebras, we hope to get some insight of how the tilted algebras are built up. By minimal non-tilted we mean a triangular algebra which is not tilted but any of its semi-convex subcategory is tilted. In this talk, we will discuss this problem and give several results of classification of classes of minimal non-tilted algebras. The results discussed here are joint work with J. A. de la Pena (UNAM-Mexico) and S. Trepode (Universidad de Mar del Plata Argentina) and are contained in [1, 2, 3].

Two Boolean algebras are elementarily equivalent if and only if they sat- isfy the same rst-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P -space.

We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C$^*$- and W$^*$-algebras; we show that interpolation space techniques can be used in the place of GNS type arguments. We define a notion of injectivity for dual Banach algebras, and show that this is equivalent to Connes-amenability. We conclude by looking at the problem of defining a well-behaved tensor product for dual Banach algebras., Comment: 40 pages; Update corrects some mathematics, and merges two sections to make for easier reading

The ordinary (or classical) Birman-Wenzl-Murakami algebras were initially conceived as an algebraic framework for the Kauffman link invariant. They also appear as centralizer algebras for representations of quantum universal enveloping algebras of orthogonal or symplectic types. It was shown by Morton and Wassermann that the BMW algebras are isomorphic to algebras of tangles in (disc $\times$ interval), modulo Kauffman skein relations. This isomorphism allows one to see very clearly certain properties of the BMW algebras -- for example existence of Markov traces and conditional expectations and a good basis. In the present work, we establish the correponding results for affine BMW algebras, showing they are isomorphic to algebras of tangles in (annulus $\times$ interval), modulo Kaufffman skein relations. Again, this provides an important foundation for studying these algebras, as the existence of Markov traces and conditional expectations are consequences of the isomorphism. Moreover, we obtain a basis of the affine BMW algebras analogous to a well known basis of the affine Hecke algebras.

Complete and σ-complete Boolean algebras of projections acting in a Banach space were introduced by W. Bade in the 1950’s. A basic fact is that every complete Boolean algebra of projections is necessarily a closed set for the strong operator topology. Here we address the analogous question for σ-complete Boolean algebras: are they always a sequentially closed set for the strong operator topology? For the atomic case the answer is shown to be affirmative. For the general case, we develop criteria which characterize when a σ-complete Boolean algebra of projections is sequentially closed. These criteria are used to show that both possibilities occur: there exist examples which are sequentially closed and others which are not (even in Hilbert space).

We construct algebras of sets which are not MB-representable. The ex- istence of such algebras was previously known under additional set-theoretic assump- tions. On the other hand, we prove that every Boolean algebra is isomorphic to an MB- representable algebra of sets.

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

We develop the representation theory of selfinjective algebras which admit Galois coverings by the repetitive algebras of algebras whose derived category of bounded complexes of nite-dimensional modules is equivalent to the derived category of coherent sheaves on a weighted projective line with virtual genus greater than one. Introduction. In this paper, by an algebra we mean a nite-dimension- al basic connected algebra (associative, with an identity) over an alge- braically closed eld K. For an algebra we denote by mod the category of nite-dimensional right -modules and by D : mod ! mod op the standard duality HomK( ;K). If all projective modules in mod are in- jective, then is called selfinjective. The classical examples of selfinjective algebras are provided by the blocks of group algebrasKG of nite groups G, or more generally by the Hopf algebras. An important class of selfinjective algebras is formed by the algebras of the form b B=G, where b B is the repetitive algebra (19) (locally nite-dimensional , without identity)

Let : M! N be a zero-product preserving linear map between algebras. We show that under some mild conditions is a product of a central element and an algebra homomorphism. Our result applies to matrix algebras, standard operator algebras, C -algebras and W -algebras. 1. Introduction. Let M and N be algebras over a eld F and : M! N a linear map. We say that is a zero-product preserving map if (a) (b) = 0 in N whenever ab = 0 in M. For example, if h is an element in the center of N and ' : M ! N is an algebra homomorphism then = h' is zero- product preserving. In this paper, we show that in many interesting cases

It was proved by Juhasz and Weiss that for every ordinal with 0 < < ! 2 there is a superatomic Boolean algebra of height and width !. We prove that if is an infinite cardinal such that < = and is an ordinal such that 0 < < ++ , then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height and width . Furthermore, iterating this forcing through all < ++ , we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic extension there is a superatomic Boolean algebra of height and width for every < ++ . Consistency for specific , like !1, then follows as a corollary.

Weak direct products of arbitrary universal algebras are introduced. The usual notion for groups and rings is a special case. Some universal algebraic properties are proved and applications to cylindric and polyadic algebras are considered.

Partitioner algebras are defined in [2] and are natural tools for studying the properties of maximal almost disjoint families of subsets of ω. In this paper we investigate which free algebras can be represented as partitioner algebras or as subalgebras of partitioner algebras. In so doing we answer a question raised in [2] by showing that the free algebra with א1 generators is represented. It was shown in [2] that it is consistent that the free Boolean algebra of size continuum is not a subalgebra of any partitioner algebra.