The quasi-strict topology on the space of quasi-multipliers of a B*-algebra

1987 ◽  
Vol 101 (3) ◽  
pp. 555-566 ◽  
Author(s):  
M. S. Kassem ◽  
K. Rowlands
Keyword(s):  
General Theory ◽  
Banach Algebra ◽  
A Priori ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Strict Topology ◽  
General Object ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Special Case

The notion of a left (right, double) multiplier may be regarded as a generalization of the concept of a multiplier to a non-commutative Banach algebra. Each of these is a special case of a more general object, namely that of a quasi-multiplier. The idea of a quasi-multiplier was first introduced by Akemann and Pedersen in ([1], §4), where they consider the quasi-multipliers of a C*-algebra. One of the defects of quasi-multipliers is that, at least a priori, there does not appear to be a way of multiplying them together. The general theory of quasi-multipliers of a Banach algebra A with an approximate identity was developed by McKennon in [5], and in particular he showed that the quasi-multipliers of a considerable class of Banach algebras could be multiplied. McKennon also introduced a locally convex topology γ on the space QM(A) of quasi-multipliers of A and derived some of the elementary properties of (QM(A), γ).

scholarly journals Automatic continuity of positive functionals on topological involution algebras

1981 ◽  
Vol 23 (2) ◽  
pp. 265-281 ◽  
Author(s):  
P.G. Dixon
Keyword(s):  
Banach Algebra ◽  
Spectral Theory ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Automatic Continuity ◽  
Topological Algebras ◽  
Positive Functional ◽  
Open Questions ◽  
Positive Functionals ◽  
Locally Convex

This paper surveys the known results on automatic continuity of positive functionals on topological *-algebras and then shows how two theorems on Banach *-algebras extend to complete metrizable topological *-algebras. The two theorems concerned are Loy's theorem on separable Banach *-algebras A with centre Z such that AZ is of countable codimension and Varopoulos' result on Banach *-algebras with bounded approximate identity. Both theorems have the conclusion that all positive functionals on such algebras are continuous. The extension of the second theorem requires the algebra to be locally convex and the approximate identity to be ‘uniformly bounded’. Neither extension requires the algebra to be LMC. This means that the proof of the first theorem is quite different from the corresponding Banach algebra result (which used spectral theory). The proof of the second is closer to the previously known LMC version, but actually neater by being more general. It is also shown that the well-known estimate of |f(a*ba)| for a positive functional f on a Banach *-algebra may be obtained without the usual use of spectral theory. The paper concludes with a list of open questions.

Amenability of Banach algebras

1989 ◽  
Vol 105 (2) ◽  
pp. 351-355 ◽  
Author(s):  
Frédéric Gourdeau
Keyword(s):  
Banach Algebra ◽  
Metric Space ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Commutative Banach Algebra ◽  
Cohomology Theory ◽  
Compact Metric Space ◽  
Amenable Banach Algebra ◽  
Second Dual

We consider the problem of amenability for a commutative Banach algebra. The question of amenability for a Banach algebra was first studied by B. E. Johnson in 1972, in [5]. The most recent contributions, to our knowledge, are papers by Bade, Curtis and Dales [1], and by Curtis and Loy [3]. In the first, amenability for Lipschitz algebras on a compact metric space K is studied. Using the fact, which they prove, that LipαK is isometrically isomorphic to the second dual of lipαK, for 0 < α < 1, they show that lipαK is not amenable when K is infinite and 0 < α < 1. In the second paper, the authors prove, without using any serious cohomology theory, some results proved earlier by Khelemskii and Scheinberg [8] using cohomology. They also discuss the amenability of Lipschitz algebras, using the result that a weakly complemented closed two-sided ideal in an amenable Banach algebra has a bounded approximate identity. Their result is stronger than that of [1].

Weak Amenability of a Class of Banach Algebras

2001 ◽  
Vol 44 (4) ◽  
pp. 504-508 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Dual Algebra ◽  
Weak Amenability ◽  
Second Dual

AbstractWe show that, if a Banach algebra is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of implies the (2m+ 1)-weak amenability of for all m ≥ 1.

scholarly journals Factorization and bounded approximate identities for a class of convolution Banach algebras

1986 ◽  
Vol 28 (2) ◽  
pp. 211-214 ◽  
Author(s):  
S. I. Ouzomgi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Approximate Identities

An algebra A factors if, for each a ∈ A, there exist b, c ∈ A with a = bc. A bounded approximate identity for a Banach algebra A is a net (eα) ⊂ A such that aeα → a and eαa → a for each a ∈ A and such that sup ‖eα ‖ < ∞. It is well known [2, 11.10] that if A has a bounded approximate identity, then A factors. But a Banach algebra may factor even if it does not have a bounded approximate identity: an example which is non-commutative and separable, and an example which is commutative and nonseparable, are given in [3, §22]. However, we do not know an example of a commutative, separable Banach algebra which factors, but which does not have a bounded approximate identity. See 4 for related work.

Factorization and unbounded approximate identities in Banach algebras

1990 ◽  
Vol 107 (3) ◽  
pp. 557-571 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Factorization Theorem ◽  
Basic Form ◽  
Converse Result ◽  
Approximate Identities

Cohen's Factorization Theorem says, in its basic form, that if A is a Banach algebra with a bounded left approximate identity, then every element x ∈ A may be written as a product x = ay for some a, y ∈ A. Such is the beauty and importance of this result that much interest attaches to the question of whether the hypothesis of a bounded left approximate identity can be weakened, or whether a converse result exists. This paper contributes to the study of that question.

scholarly journals Approximate complementation and its applications in studying ideals of Banach algebras

2003 ◽  
Vol 92 (2) ◽  
pp. 301 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Compact Subset ◽  
Approximation Property ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Amenable Banach Algebra

We show that a subspace of a Banach space having the approximation property inherits this property if and only if it is approximately complemented in the space. For an amenable Banach algebra a closed left, right or two-sided ideal admits a bounded right, left or two-sided approximate identity if and only if it is bounded approximately complemented in the algebra. If an amenable Banach algebra has a symmetric diagonal, then a closed left (right) ideal $J$ has a right (resp. left) approximate identity $(p_{\alpha})$ such that, for every compact subset $K$ of $J$, the net $(a\cdot p_{\alpha})$ (resp. $(p_{\alpha}\cdot a)$) converges to $a$ uniformly for $a \in K$ if and only if $J$ is approximately complemented in the algebra.

Isomorphisms and multipliers on second dual algebras of Banach algebras

1992 ◽  
Vol 111 (1) ◽  
pp. 161-168 ◽  
Author(s):  
Fereidoun Ghahramani ◽  
Anthony To-Ming Lau
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Weakly Compact ◽  
Dual Algebra ◽  
Arens Product ◽  
Second Dual

Suppose that A is a Banach algebra and let A be the second dual algebra of A equipped with the first Arens product 3. In this paper we characterize compact and weakly compact multipliers of A, when A possesses a bounded approximate identity and is a two sided ideal in A. We use this to study the isomorphisms between second duals of various classes of Banach algebras satisfying the above properties.

On The Center Of Quasi-Central Banach Algebras With bounded Approximate Identity

1981 ◽  
Vol 33 (1) ◽  
pp. 68-90
Author(s):  
Sin-Ei Takahasi
Keyword(s):  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Central Complex ◽  
Double Centralizer ◽  
Completely Regular ◽  
Continuous Complex ◽  
Complex Valued ◽  
Canonical Image

Let A be a quasi-central complex Banach algebra with a bounded approximate identity and Prim A the structure space of A. In [15], we have shown that every central double centralizer T on A can be represented as a bounded continuous complex-valued function ΦT on Prim A such that Tx + P = ΦT(P)(x + P) for all x ∈ A and P ∈ Primal when the center Z(A) of A is completely regular. Here x + P for P ∈ Prim A denotes the canonical image of x in A/P. In particular, in the case of quasi-central C*-algebras, this result is equivalent to the Dixmier's representation theorem of central double centralizers on C*-algebras (see [3, Section 2] and [9, Theorem 5]).In this paper, it is shown that if Z(A) is completely regular then the space Prim A is locally quasi-compact and for each element z of Z(A), ΦLz vanishes at infinity, where Lz for z ∊ Z(i) is the central double centralizer on A defined by Lz(x) = zx for all x ∊ A.

Spectra of approximate identities in Banach algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 271-278 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Unit Interval ◽  
Complex Banach Algebra ◽  
Approximate Identities

1. Introduction. The aim of this paper is to show that, in every complex Banach algebra with a one-sided or two-sided bounded approximate identity, there exists another bounded approximate identity of the same sort whose spectra lie close to the unit interval [0, 1].

scholarly journals Quasimultipliers on -Algebras

2011 ◽  
Vol 2011 ◽  
pp. 1-30 ◽  
Author(s):  
Marjan Adib ◽  
Abdolhamid Riazi ◽  
Liaqat Ali Khan
Keyword(s):  
Banach Algebras ◽  
Approximate Identity ◽  
Normed Algebra ◽  
Joint Continuity ◽  
Carry Over ◽  
Locally Convex

We investigate the extent to which the study of quasimultipliers can be made beyond Banach algebras. We will focus mainly on the class of -algebras, in particular on complete -normed algebras, , not necessarily locally convex. We include a few counterexamples to demonstrate that some of our results do not carry over to general -algebras. The bilinearity and joint continuity of quasimultipliers on an -algebra are obtained under the assumption of strong factorability. Further, we establish several properties of the strict and quasistrict topologies on the algebra of quasimultipliers of a complete -normed algebra having a minimal ultra-approximate identity.

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