The Banach algebra induced by a double centralizer

Given a Banach algebra , R. Larsen defined, in his book ``An introduction to the theory of multipliers", a Banach algebra by means of a multiplier on , and essentially used it in the case of a commutative semisimple Banach algebra to prove a result on multiplications which preserve regular maximal ideals. Here, we consider the analogue Banach algebra induced by a bounded double centralizer of a Banach algebra . Then, our main concern is devoted to the relationships between , , and the algebras of bounded double centralizers and of and , respectively. By removing the assumption of semisimplicity, we generalize some results proven by Larsen.

     

Zero product preserving maps of operator-valued functions

Let be locally compact Hausdorff spaces and , be Banach algebras. Let be a zero product preserving bounded linear map with dense range. We show that is given by a continuous field of algebra homomorphisms from into if is irreducible. As corollaries, such a surjective arises from an algebra homomorphism, provided that is a -algebra and is a semi-simple Banach algebra, or both and are -algebras.

     

Non-reflexivity of the derivation space from Banach algebras of analytic functions

Let be an open connected subset of the plane, and let be a Banach algebra of analytic functions on . We show that the space of bounded derivations from into is not reflexive. We also obtain similar results when for .

     

Semiprime smash products and -stable prime radicals for PI-algebras

Assume that is a finite-dimensional Hopf algebra over a field and that is an -module algebra satisfying a polynomial identity (PI). We prove that if is semisimple and is -semiprime, then is semiprime. If is cosemisimple, we show that the prime radical of is -stable.

     

On the extensions of homogeneous polynomials

We investigate the problem of the uniqueness of the extension of -homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space that admits contractive projection of finite rank of at least dimension 2, for every there exists an -homogeneous polynomial on that has infinitely many extensions to . We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice , for instance when satisfies an upper -estimate with constant one for some , any -homogeneous polynomial on attaining its norm at with a finite rank band projection , has a unique extension to its bidual . We apply these results in a class of Orlicz sequence spaces.

     

Actions of pointed Hopf algebras with reduced pi invariants

Let be an -module algebra, where is a pointed Hopf algebra acting on finitely of dimension . Suppose that for every nonzero -stable left ideal of . It is proved that if satisfies a polynomial identity of degree , then satisfies a polynomial identity of degree provided at least one of the following additional conditions is fulfilled:

1. is semiprime and is almost central in ,
2. is reduced.

If we also assume that is central, then satisfies the standard polynomial identity of degree , where is the greatest integer in .

     

A structure theorem for quasi-Hopf comodule algebras

If is a quasi-Hopf algebra and is a right -comodule algebra such that there exists a morphism of right -comodule algebras, we prove that there exists a left -module algebra such that . The main difference when comparing to the Hopf case is that, from the multiplication of , which is associative, we have to obtain the multiplication of , which in general is not; for this we use a canonical projection arising from the fact that becomes a quasi-Hopf -bimodule.

     

Subsemivarieties of -algebras

A variety is a class of Banach algebras , for which there exists a family of laws such that is precisely the class of all Banach algebras which satisfies all of the laws (i.e. for all , . We say that is an -variety if all of the laws are homogeneous. A semivariety is a class of Banach algebras , for which there exists a family of homogeneous laws such that is precisely the class of all Banach algebras , for which there exists 0$"> such that for all homogeneous polynomials , , where . However, there is no variety between the variety of all -algebras and the variety of all -algebras, which can be defined by homogeneous laws alone. So the theory of semivarieties and the theory of varieties differ significantly. In this paper we shall construct uncountable chains and antichains of semivarieties which are not varieties.

     

Sufficient conditions for a linear functional to be multiplicative

A commutative Banach algebra is said to have the property if the following holds: Let be a closed subspace of finite codimension such that, for every , the Gelfand transform has at least distinct zeros in , the maximal ideal space of . Then there exists a subset of of cardinality such that vanishes on , the set of common zeros of . In this paper we show that if is compact and nowhere dense, then , the uniform closure of the space of rational functions with poles off , has the property for all . We also investigate the property for the algebra of real continuous functions on a compact Hausdorff space.

     

Hahn-Banach operators

We consider real spaces only.

Definition. An operator between Banach spaces and is called a Hahn-Banach operator if for every isometric embedding of the space into a Banach space there exists a norm-preserving extension of to .

A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces such that there exists a Hahn-Banach operator of rank . The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman.

     

A structure of ring homomorphisms on commutative Banach algebras

We give a structure theorem for a ring homomorphism of a commutative regular Banach algebra into another commutative Banach algebra. In particular, it is shown that:

(i)

A ring homomorphism of a commutative -algebra onto another commutative -algebra with connected infinite Gelfand space is either linear or anti-linear.

(ii)

A ring automorphism of is either linear or anti-linear.

(iii)

, and the disc algebra are neither ring homomorphic images of nor .

     

Elements in a commutative Banach algebra determining the norm topology

For an element of a commutative complex Banach algebra we investigate the following property: every complete norm on making the multiplication by from to itself continuous is equivalent to .

     

Extensions of orthosymmetric lattice bimorphisms revisited

This note furnishes an example illustrating the following two facts. On the one hand, there exist Archimedean Riesz spaces and with Dedekind-complete and an orthosymmetric lattice bimorphism with lattice bimorphism extension which is not orthosymmetric, where denotes the Dedekind-completion of . On the other hand, there is an associative -multiplication in the same Archimedean Riesz space which extends to a -multiplication in which is not associative. The existence of such an example provides counterexamples to assertions in Toumi, 2005.

     

Imbeddings of free actions on handlebodies

Fix a free, orientation-preserving action of a finite group on a -dimensional handlebody . Whenever acts freely preserving orientation on a connected -manifold , there is a -equivariant imbedding of into . There are choices of closed and Seifert-fibered for which the image of is a handlebody of a Heegaard splitting of . Provided that the genus of is at least , there are similar choices with closed and hyperbolic.

     

Invariant projections and convolution operators

We prove the existence of invariant projections from the Banach space of -pseudomeasures onto with for closed neutral subgroup of a locally compact group . As a main application we obtain that every closed neutral subgroup is a set of -synthesis in and in fact locally -Ditkin in . We also obtain an extension theorem concerning the Fourier algebra.

     

Quasitilted extensions of algebras I

Let be a connected finite dimensional -algebra, and let be a nonzero decomposable -module such that the one-point extension is quasitilted. We show here that every nonzero indecomposable direct summand of is directing and is a tilted algebra.

     

A group structure on squares

We show that there is an abelian group structure on the orbit set of ``squares' of unimodular rows of length over a commutative ring of stable dimension when , odd and also an abelian group structure on the orbit set of ``fourth powers' of unimodular rows of length over a commutative ring of stable dimension when , even.

     

Normal bases for Hopf-Galois algebras

Let be a Hopf algebra over a commutative ring such that is a finitely generated, projective module over , let be a right -comodule algebra, and let be the subalgebra of -coinvariant elements of . If is a Galois extension of and is a local subalgebra of the center of , then is a cleft right -comodule algebra or, equivalently, there is a normal basis for over .

     

A reduction theorem for the topological degree for mappings of class

The reduction theorem for the Leray-Schauder degree provides an efficient tool to calculate the value of the degree in a suitable invariant subspace. We shall prove how the calculation of the value of the topological degree for a mapping of class from a real separable reflexive Banach space into the dual space can be reduced into the calculation of degree of mapping from a closed subspace into Since the Leray-Schauder mappings are acting from to and we are dealing with mappings from to the standard `invariant subspace' condition must be replaced by a less obvious one.

     

Domination by positive disjointly strictly singular operators

We prove that each positive operator from a Banach lattice to a Banach lattice with a disjointly strictly singular majorant is itself disjointly strictly singular provided the norm on is order continuous. We prove as well that if is dominated by a disjointly strictly singular operator, then is disjointly strictly singular.