Polynomially Riesz elements

Abstract

A Banach algebra element a ∈ A is said to be “polynomially Riesz”, relative to the homomorphism T : A → B, if there exists a nonzero complex polynomial p(z) such that the image T p(a) ∈ B is quasinilpotent.     

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.     

ON INESSENTIAL IDEALS IN BANACH ALGEBRAS

Abstract

In [2], Aupetit studied the perturbation of elements of a Banach algebra A by elements of an inessential ideal I of A. The main result of his paper is based on a lemma ([2], theorem 1.1) obtained by the use of subharmonic methods and analytic multivalued functions. Our aim in this note is to prove Auptetit's perturbation theorem ([2], theorem 2.4) by making use of elementary methods.     

Cohomological Characterizations of Biprojective and Biflat Banach Algebras

 Let A be a biprojective Banach algebra, and let A-mod-A be the category of Banach A-bimodules. In this paper, for every given -mod-A, we compute all the cohomology groups . Furthermore, we give some cohomological characterizations of biprojective Banach algebras. In particular, we show that the following properties of a Banach algebra A are equivalent to its biprojectivity: (i) for all -mod -A; (ii) for all -mod-A; (iii) for all -mod-A. (Here and are, respectively, the Banach A-bimodules of left, right and double multipliers of X.) Further, if A is a biflat Banach algebra and -mod-A, we compute all the cohomology groups , where is the Banach A-bimodule dual to X. Also, we give cohomological characterizations of biflat Banach algebras. We prove that a Banach algebra A is biflat if and only if any of the following conditions is valid: (i’) for all -mod-A; (ii’) for all -mod-A; (iii’) for all -mod-A. Received 16 June 1998     

QUOTIENTS OF 2-SUMMING ALGEBRAS

Abstract

A result of A.M. Mantero and A. Tonge is strengthened by giving an example of a commutative operator algebra which is semisimple but which is not the quotient algebra of a 2-summing Banach algebra by a closed ideal.     

Quotient rings of algebras of functions and operators

Abstract. Various quotient rings of rings B of Banach algebra A-valued continuous functions on a completely regular Hausdorff Space X are constructed in terms of continuous functions defined on dense open subsets of X taking values in the maximal quotient ring of the Banach algebra A. This extends the results proved by N. J. Fine, L., Gillman and J. Lambek (1965) for the case of A, the field of real numbers. The pattern is similar and utilizes as well as generalises the results proved for algebras of multipliers of B by C. A. Akemann, G. K. Pedersen and J. Tomiyama (1973). The techniques combine those from algebra, analysis and topology. The details of the cases when A is the normed division algebra of real quaternions or the operator algebra B(H) of a Hilbert space H are given to illustrate our results. Received February 1, 1999; in final form June 18, 1999 / Published online May 8, 2000     

Automatic continuity of n-homomorphisms between FrÉchet algebras

We impose a condition on a commutative regular Fréchet algebra (A, (p_m )) to ensure that A/kerp_m is a Fréchet Q-algebra. This implies that if θ is an n-homomorphism on certain Fréchet algebras (A, (p_m )) into semisimple commutative Fréchet algebras (B,(q_m)) such that θ(kerp_m) ? kerq_m, for large enough m, then θ is continuous. We also show that if A is a Fréchet Q-algebra, B is a semisimple Fréchet algebra, θ: A → B is a dense range n-homomorphism such that θ(A) is factorizable, and the spectral radius v_B is continuous on the separating space (θ), then θ is automatically continuous.     

A MULTIPLICATION INEQUALITY IN COMPLEX BANACH LATTICE ALGEBRAS

Abstract

We present an alternative order theoretic proof for the fundamental result that |Z₁, Z₂ | ? |Z₁ 1 | Z₂ | with z₂ and z₂ elements of the complexification of an Archimedean uniformly complete Riesz algebra.     

SECOND-ORDER BOOLEAN ALGEBRAS

Opgedra aan Prof. Hennie Schutte by geleentheid van sy sestigste verjaarsdag.

Abstract

A Boolean algebra is the algebraic version of a field of sets. The complex algebra C(B) of a Boolean algebra B is defined over the power set of B; it is a field of sets with extra operations. The notion of a second-order Boolean algebra is intended to be the algebraic version of the complex algebra of a Boolean algebra. To this end a representation theorem is proved.     

SPECTRAL PROPERTIES OF POSITIVE ELEMENTS IN BANACH LATTICE ALGEBRAS

Abstract

We prove that if a unital Banach lattice algebra has sufficiently many one-dimensional elements and if its unit element has sufficiently many components then its positive elements have spectral properties analogous to those of positive operators on Banach lattices. In particular, if a positive element is irreducible (in the sense that (1—e)xe > 0 for all components e of 1 satisfying 0 ≠ e ≠ 1) and compact, its spectral radius is positive and its spectrum shows cyclic behaviour.     

Spectral mapping theorems for the upper Weyl and upper Browder spectra

Abstract

We discuss the spectral mapping properties of the upper Weyl and upper Browder spectra of an arbitrary ordered Banach algebra element.     

Zero-separating algebras of continuous functions

Let A be a lattice-ordered algebra endowed with a topology compatible with the structure of algebra. We provide internal conditions for A to be isomorphic as lattice-ordered algebras and homeomorphic to Ck(X), the lattice-ordered algebra C(X) of real continuous functions on a completely regular and Hausdorff topological space X, endowed with the topology of uniform convergence on compact sets. As a previous step, we determine this topology among the locally m-convex topologies on C(X) with the property that each order closed interval is bounded.     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

Lattice and Algebra Homomorphisms on C(X) in Zermelo-Fraenkel Set Theory

Abstract

Let C (X) denote the lattice-ordered algebra of all real-valued continuous functions on a topological space X. This paper discusses in Zermelo-Fraenkel Set Theory the equivalence on C (X) between algebra homomorphisms, lattice homo- morphisms, and point evaluations.     

Commutatively ordered Banach algebras

Abstract

Spectral theory in ordered Banach algebras (OBAs) has been investigated and several authors have made contributions. However, the results are not applicable to non-commutative C^*-algebras, since a non-commutative C^*-algebra is not an OBA. In this paper we introduce a more general structure, called a commutatively ordered Banach algebra (COBA), which includes the class of OBAs. Every C^* - algebra is a COBA. We will give the basic properties of COBAs and show how known results in OBAs can be generalized to the COBA setting. We will then discuss two spectral problems regarding COBA elements. The results obtained, of course, hold true in an OBA as well. These results extend the theory of COBAs and OBAs.     

Additive results for the generalized Drazin inverse in a Banach algebra

In this paper we investigate additive properties of the generalized Drazin inverse in a Banach algebra. We find some new conditions under which the generalized Drazin inverse of the sum a + b could be explicitly expressed in terms of a, a^d, b, b^d. Also, some recent results of Castro and Koliha [New additive results for the g-Drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 1085-1097] are extended.     

Additivity of Jordan maps on standard Jordan operator algebras

Let A be a standard Jordan operator algebra on a Hilbert space of dimension >1 and B be an arbitrary Jordan algebra. In this note, we prove that if a bijection ?:A→B satisfies

     

Symmetric topological*-algebras

V. Pták's inequality is valid for every hermitian completeQ locallym-convex (:l.m.c.) algebra. Every algebra of the last kind is, in particular, symmetric. Besides, a (Hausdorff) locallyC ^*-algebra (being always symmetric) with the propertyQ is, within a topological algebraic isomorphism, aC ^*-algebra. Furthermore, a type of Raikov's criterion for symmetry is also valid for non-normed topological^*-algebras. Concerning topological tensor products, one gets that symmetry of the-completed tensor product of two unital Fréchet l.m.c.^*-algebrasE, F ( denotes the projective tensorial topology) is always passed toE, F, while the converse occurs when moreover either ofE, F is commutative.     

Radius preserving (semi)regularities in Banach algebras

Abstract

In this note we investigate regularities (semiregularities) R and S in a Banach algebra A satisfying S ? R and the corresponding spectra σ_S and σ_R Satisfying

sup {|λ| : λ ∈ σ_R(a)} = sup{|λ| : λ ∈ σ_S (a)}     

Generalized inverses over Banach algebras

LetA be a matrix over a complex commutative unital Banach algebra. We give necessary and sufficient conditions forA to have a generalized inverse. Moreover, if the Banach algebra has a symmetric involution, these are also necessary and sufficient conditions forA to admit the Moore-Penrose inverse.Partially supported by NSF Grant DMS-8802593