σ-Contractible and σ-biprojective Banach algebras

Abstract

The notion of σ-amenability for Banach algebras and its related notions were introduced and extensively studied by M.S. Moslehian and A.N. Motlagh in [10]. We develop these notions parallel to the amenability of Banach algebras introduced by B.E. Johnson in [5]. Briefly, we investigate σ-contractibility and σ-biprojectivity of Banach algebras, which are extensions of usual notions of contractibility and biprojectivity, respectively, where σ is a bounded endomorphism of the corresponding Banach algebra. We also give the notion σ-diagonal. Then we verify relations between σ-contractibility, σ-biprojectivity and the existence of a σ-diagonal for a Banach algebra, when σ has dense range or is an idempotent. Moreover, we obtain some hereditary properties of these concepts.     

Normed algebras with involution

We show that most of the theory of Hermitian Banach algebras can be proved for normed *-algebras without the assumption of completeness. The conditionr(x)≤p(x) for allx (wherep(x)=r(x ^* x) ^1/2 is the Pták function), which is essential in the theory of Hermitian Banach algebras, is replaced for normed *-algebras by the conditionr(x+y)≤p(x)+p(y) for allx, y. In case of Banach *-algebras these conditions are equivalent. The research has been supported by a grant from La Junta de Andalucía and by the Department of Applied Mathematics, University of Seville This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

Some applications of Banach algebra techniques

We consider some functional Banach algebras with multiplications as the usual convolution product * and the so‐called Duhamel product ?. We study the structure of generators of the Banach algebras (C⁽ⁿ⁾[0, 1], *) and (C⁽ⁿ⁾[0, 1], ?). We also use the Banach algebra techniques in the calculation of spectral multiplicities and extended eigenvectors of some operators. Moreover, we give in terms of extended eigenvectors a new characterization of a special class of composition operators acting in the Lebesgue space L^p[0, 1] by the formula (C_φf)(x) = f(φ(x)).     

Ideal amenability of Banach algebras on locally compact groups

In this paper we study the ideal amenability of Banach algebras. LetA be a Banach algebra and letI be a closed two-sided ideal inA, A isI-weakly amenable ifH ¹(A,I ^*) = {0}. Further,A is ideally amenable ifA isI-weakly amenable for every closed two-sided idealI inA. We know that a continuous homomorphic image of an amenable Banach algebra is again amenable. We show that for ideal amenability the homomorphism property for suitable direct summands is true similar to weak amenability and we apply this result for ideal amenability of Banach algebras on locally compact groups.     

Homomorphisms of certain Banach function algebras

In this note, we study homomorphisms with domainD ⁿ(X) orLipα(X, d) of which ranges are certain Banach function algebras and determine in which cases these homomorphisms are compact.     

Some remarks on derivations in Banach algebras and related results

Summary In the first section of this paper we consider some functional equations which are closely connected to derivations (i.e. additive mappings with the propertyD(ab) = aD(b) + D(a)b) on Banach algebras. IfD is a derivation on some algebraA, then the equationD(a) = – aD(a ^–1 )a holds for all invertible elementsa A. It seems natural to ask whether this functional equation characterizes derivations among all additive mappings. It is too much to expect an affirmative answer to this question in arbitrary algebras, since it may happen that even in normed algebras the group of all invertible elements contains only scalar multiples of the identity. We try to answer the question above in Banach algebras, since in Banach algebras invertible elements exist in abundance. In the second section of the paper we prove some results concerning representability of quadratic forms by bilinear forms.     

Ditkin's theorem and [SIN]-groups

A general Ditkin-type theorem is proved for certain Banach algebras. Applications of this result are derived concerningL ¹ (G) whereG is a [SIN]-group.This research was partially supported by NSF Grant MCS 8002525.     

On the Generalized Drazin Inverse and Generalized Resolvent

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in >C ^*-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, 2 × 2 operator matrices are considered. As corollaries, we get some well-known results.     

The Banach algebras generated by representations of abelian semigroups

Let S be a suitable subsemigroup of a locally compact abelian group. We consider the class of Banach algebras A for which there exists a continuous homomorphism ω : L ¹ (S) → A with dense range. We study the behavior of the radical in such algebras. As an application, we present some results concerning the structure of weakly compact homomorphisms of semigroup algebras.     

On a valuation field invented by A. Robinson and certain structures connected with it

We clarify the structure of the non-archimedean valuation field ^ρ R which was introduced by A. Robinson, and of theρ-non-archimedean hulls of Banach algebras and Lie groups. (For Banach spaces this construction is due to W. A. J. Luxemburg.) In particular, we show that any two infinite-dimensional real normed spaces have a pair of isometrically isomorphicρ-non-archimedean hulls. Dedicated to the memory of A. Robinson on the occasion of the 70th anniversary of his birth.     

Linear Functional Equations in Banach Modules over a C *-Algebra

The paper is a survey on the Hyers–Ulam–Rassias stability of linear functional equations in Banach modules over a C ^*-algebra. Its contents is divided into the following sections: 1. Introduction; 2. Stability of the Cauchy functional equation in Banach modules; 3. Stability of the Jensen functional equation in Banach modules; 4. Stability of the Trif functional equation in Banach modules; 5. Stability of cyclic functional equations in Banach modules over a C ^*-algebra; 6. Stability of cyclic functional equations in Banach algebras and approximate algebra homomorphisms; 7. Stability of algebra *-homomorphisms between Banach *-algebras and applications.     

Generalized stable ranks for commutative Banach algebras

We introduce the notion of generalized E-stable ranks for commutative unital Banach algebras and determine these ranks for the disk-algebra A(\mathbbD){A(\mathbb{D})}, many of its subalgebras, and the algebra H ^∞ of bounded holomorphic functions in the unit disk. Relations to L-sets and separating algebras, notions due to Csordas and Reiter, are given, too. Finally we show that the absolute stable rank of A(\mathbbD){A(\mathbb{D})} and H ^∞ is bigger than 2.     

New Holomorphically Closed Subalgebras of C*-Algebras of Hyperbolic Groups

We present a new construction of dense, isospectral subalgebras of unconditional Banach algebras over word-hyperbolic groups. We study the algebras thus obtained and derive applications to delocalized L ²-invariants of closed Riemannian manifolds of negative curvature and to the local cyclic cohomology of the reduced group C*-algebras of word-hyperbolic groups.     

Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If \frak A {\frak A} is a reflexive, amenable Banach algebra such that for each maximal left ideal L of \frak A {\frak A} (i) the quotient \frak A/L {\frak A}/L has the approximation property and (ii) the canonical map from \frak A \check? L^{^} {\frak A} \check{\otimes} L^\perp to (\frak A / L) \check? L^{^} ({\frak A} / L) \check{\otimes} L^\perp is open, then \frak A {\frak A} is finite-dimensional. As an application, we show that, if \frak A {\frak A} is an amenable Banach algebra whose underlying Banach space is an \scr L^p {\scr L}^p -space with p ? (1,￥) p\in (1,\infty) such that for each maximal left ideal L the quotient \frak A/L {\frak A}/L has the approximation property, then \frak A {\frak A} is finite-dimensional.     

A non-reflexive Grothendieck space that does not containl ∞

A compact spaceS is constructed such that, in the dual Banach spaceC(S)^*, every weak^* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol _∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl _∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.     

Hilbert-Carleman and regularized determinants for linear operators

A general theory of regularized and Hilbert-Carleman determinants in normed algebras of operators acting in Banach spaces is proposed. In this approach regularized determinants are defined as continuous extensions of the corresponding determinants of finite dimensional operators. We characterize the algebras for which such extensions exist, describe the main properties of the extended determinants, obtain Cramer's rule and the formulas for the resolvent which are expressed via the extended tracestr(A ^k ) of iterations and regularized determinants.This paper is a continuation of the paper [GGKr].     

Derivations into duals of ideals of Banach algebras

We introduce two notions of amenability for a Banach algebra A. LetI be a closed two-sided ideal inA, we sayA is I-weakly amenable if the first cohomology group ofA with coefficients in the dual space I* is zero; i.e.,H ¹(A, I*) = {0}, and,A is ideally amenable ifA isI-weakly amenable for every closed two-sided idealI inA. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study theI-weak amenability of a Banach algebraA for some special closed two-sided idealI.     

Character Connes amenability of dual Banach algebras

We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A^** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.     

Amenability constants for semilattice algebras

For any finite commutative idempotent semigroup S, a semilattice, we show how to compute the amenability constant of its semigroup algebra ℓ ¹(S). This amenability constant is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We also give example of a commutative Clifford semigroups G _n whose semigroup algebras ℓ ¹(G _n ) admit amenability constants of the form 41+4(n−1)/n. We also show there is no commutative semigroup whose semigroup algebra has an amenability constant between 5 and 9. N. Spronk’s research was supported by NSERC Grant 312515-05.