Some results about the spectrum of commutative Banach algebras under the weak topology and applications

LetA be a commutative Banach algebra with a nonempty spectrum A. By weak we denote the relative weak topology induced on A by (A ^*,A ^**). In this note we study some properties of the topological space (A, weak) and present some applications of the results obtained and tools used to amenability, weakly compact homomorphisms, weakly compact subsets of the spectrum of the uniform algebras and to a characterization of the synthesizable ideals of the algebraA.     

On branching index of algebraic extension spectrum points

The paper studies the branching A-index and the topological branching index of points of the spectrum of Arens-Hoffman algebraic extension of a semisimple commutative Banach algebras A.     

Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras

The additive(generalized)ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive(generalized)ξ-Lie derivation with ξ = 1 if and only if it is an additive(generalized)derivation satisfying L(ξA)= ξL(A)for all A. These results are then used to characterize additive(generalized)ξ-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.     

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.     

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.     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

Isomorphisms between C-ternary algebras

In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in C^∗-ternary algebras and of derivations on C^∗-ternary algebras for the following Cauchy-Jensen additive mappings:

(0.1)     

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     

Inner derivations on ultraprime normed real algebras

We show that the set of all inner derivations of an ultraprime real Banach algebra is closed within all bounded derivations. More concretely, we show that for such an algebra A there exists a positive number γ (depending only on the “constant of ultraprimeness” of A) satisfying γ ∥ a+Z(A) ∥≦∥ D _a ∥ for all a in A, where Z(A) denotes the centre of A and D _a denotes the inner derivation on A induced by a. This result is an extension of the corresponding complex version obtained by the authors in [Proc. Amer. Math. Soc., to appear]. The proof relies on the following theorem: ultraproducts of a family of central ultraprime real Banach algebras with a unit and with constant of ultraprimeness greater than or equal to a fixed positive constant K are central ultraprime Banach algebras with a unit. This fact is obained via a general result for real Banach algebras that reads as follows: If A is a central real Banach algebra with a unit 1, then for every a in A satisfying ∥ 1+a ² ∥<1 we have [1+√1?||1+1a ²||]²≦2(|?l+M _a ||+||D _a ||) where M _a denotes the two-sided multiplication operator by a on A.     

Inclusion-exclusion: Exact and approximate

It is often required to find the probability of the union of givenn eventsA ₁ ,...,A _n . The answer is provided, of course, by the inclusion-exclusion formula: Pr(A _i )= _i – _i<j Pr(A _i A _j )±.... Unfortunately, this formula has exponentially many terms, and only rarely does one manage to carry out the exact calculation. From a computational point of view, finding the probability of the union is an intractable, #P-hard problem, even in very restricted cases. This state of affairs makes it reasonable to seek approximate solutions that are computationally feasible. Attempts to find such approximate solutions have a long history starting already with Boole [1]. A recent step in this direction was taken by Linial and Nisan [4] who sought approximations to the probability of the union, given the probabilities of allj-wise intersections of the events forj=1,...k. The developed a method to approximate Pr(A _i ), from the above data with an additive error of exp . In the present article we develop an expression that can be computed in polynomial time, that, given the sums _|S|=j Pr( _iS A _i ) forj=1,...k, approximates Pr(A _i ) with an additive error of exp . This error is optimal, up to the logarithmic factor implicit in the notation.The problem of enumerating satisfying assignments of a boolean formula in DNF formF=v _l ^m C _i is an instance of the general problem that had been extensively studied [7]. HereA _i is the set of assignments that satisfyC _i , and Pr( _iS A _i )=a _S /2ⁿ where _iS C _i is satisfied bya _S assignments. Judging from the general results, it is hard to expect a decent approximation ofF's number of satisfying assignments, without knowledge of the numbersa _S for, say, all cardinalities . Quite surprisingly, already the numbersa _S over |S|log(n+1)uniquely determine the number of satisfying assignments for F.We point out a connection between our work and the edge-reconstruction conjecture. Finally we discuss other special instances of the problem, e.g., computing permanents of 0,1 matrices, evaluating chromatic polynomials of graphs and for families of events whose VC dimension is bounded.Work supported in part by a grant of the Binational Israel-US Science Foundation.Work supported in part by a grant of the Binational Israel-US Science Foundation and by the Israel Science Foundation.     

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)}     

Über Positive Resolventenwerte Positiver Operatoren

In this paper we study the positive resolvent values of positive operators respectively of positive elements in Banach lattice ordered algebras. In the matrix case these values are just the inverse M-matrices. One of the main results is the following: Let A be a Banach lattice ordered algebra. A positive invertible element xA is a resolvent value of a positive element yA if and only if the element x satisfies the negative principle: If aA, < 0 and xaa then xa 0.     

New characterizations for core inverses in rings with involution

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Raki?, N. ?. Din?i? and D. S. Djordjevi? generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R. Then a ∈ R# with a# = b if and only if (ab)* = ab, ba² = a, and ab² = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.     

Krivine's theorem and the indices of a Banach lattice

In this paper we shall present an exposition of a fundamental result due to J.L. Krivine about the local structure of a Banach lattice. In [3] Krivine proved that _p (1p) is finitely lattice representable in any infinite dimensional Banach lattice. At the end of the introduction of [3] it is then stated that a value of p for which this holds is given by, what we will call below, the upper index of the Banach lattice. He states that this follows from the methods of his paper and of the paper [5] of Maurey and Pisier. One can ask whether the theorem also holds for p equal to the lower index of the Banach lattice. At first glance this is not obvious from [3], since many theorems in [3] have as a hypothesis that the upper index of the Banach lattice is finite. This can e.g. also be seen from the book [6] of H.U. Schwarz, where only the result for the upper index is stated, while both indices are discussed. One purpose of this paper is clarify this point and to present an exposition of all the ingredients of a proof of Krivine's theorem for both the upper and lower index of a Banach lattice. We first gather some definitions and state some properties of the indices of a Banach lattice. For a discussion of these indices we refer to the book of Zaanen[7].     

Projective spaces of aC *-algebra

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C^*-algebraA with a fixed projectionp. The resulting spaceP(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group ofA. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean-in Schwarz-Zaks terminology) are considered, allowing a comparison amongP(p), the Grassmann manifold ofA and the space of positive elements which are unitary with respect to the bilinear form induced by the reflection =2p–1. Among several metrical results, we prove that geodesics are unique and of minimal length when measured with the spherical and non-Euclidean metrics.Partially supported by UBACYT TW49 and TX92, PIP 4463 (CONICET) and ANPCYT PICT 97-2259 (Argentina)