scholarly journals Automatic continuity for Banach algebras with finite-dimensional radical

2006 ◽  
Vol 81 (2) ◽  
pp. 279-296 ◽  
Author(s):  
Hung Le Pham
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Automatic Continuity ◽  
Algebraic Condition ◽  
Finite Dimensional ◽  
Complete Algebra

AbstractThe paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.

Automatic Continuity of Homomorphisms in Non-associative Banach Algebras

2013 ◽  
Vol 65 (5) ◽  
pp. 989-1004
Author(s):  
C-H. Chu ◽  
M. V. Velasco
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Surjective Homomorphism ◽  
Automatic Continuity ◽  
Normed Algebra ◽  
Associative Algebras ◽  
Rare Element ◽  
Rare Elements

AbstractWe introduce the concept of a rare element in a non-associative normed algebra and show that the existence of such an element is the only obstruction to continuity of a surjective homomorphism from a non-associative Banach algebra to a unital normed algebra with simple completion. Unital associative algebras do not admit any rare elements, and hence automatic continuity holds.

Banach algebras satisfying certain chain conditions on closed ideals

2020 ◽  
Vol 57 (3) ◽  
pp. 290-297
Author(s):  
Abdullah Alahmari ◽  
Falih A. Aldosray ◽  
Mohamed Mabrouk
Keyword(s):  
Banach Algebra ◽  
Jacobson Radical ◽  
Banach Algebras ◽  
Chain Condition ◽  
Chain Conditions ◽  
Finite Dimensional ◽  
Descending Chain Condition ◽  
Unital Banach Algebra ◽  
Ascending Chain Condition

AbstractLet 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

scholarly journals Topologically simple Banach algebras with derivation

1999 ◽  
Vol 60 (1) ◽  
pp. 153-161
Author(s):  
El Hossein Illoussamen ◽  
Volker Runde
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Pivotal Role ◽  
Automatic Continuity

It is not known if a commutative, topologically simple, radical Banach algebra exists. If, however, every derivation on such an algebra is continuous, this yields the automatic continuity of all derivations on commutative, semiprime Banach algebras. Utilising techniques used by Thomas in his proof of the Singer-Wermer conjecture, we show that, if A is a commutative, topologically simple Banach algebra with a non-zero derivation on it, then a quotient of a certain localisation of A has a power series structure. A pivotal role is played by what we call ample sets of denominators.

scholarly journals On one-sided primitivity of Banach algebras

2010 ◽  
Vol 53 (1) ◽  
pp. 111-123 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Semigroup Algebra ◽  
Alternative Proof ◽  
Infinite Dimensional ◽  
Finite Dimensional

AbstractLet S be the semigroup with identity, generated by x and y, subject to y being invertible and yx = xy2. We study two Banach algebra completions of the semigroup algebra ℂS. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that ℂS is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for ℂS is finite dimensional and hence that ℂS has a separating family of such modules.

Automatic continuity with application to C*-algebras

1990 ◽  
Vol 107 (2) ◽  
pp. 345-347 ◽  
Author(s):  
Angel Rodriguez Palacios
Keyword(s):  
Banach Algebras ◽  
Direct Consequence ◽  
Automatic Continuity ◽  
Associative Algebras ◽  
C Algebras ◽  
Complete Algebra ◽  
Usual Norm

The fact proved by Cleveland [4], that the topology of any (non-complete) algebra norm on a C*-algebra is stronger than the topology of the usual norm, is reencountered as a direct consequence of a theorem, which we prove in this note, stating that homomorphisms from certain non-complete normed (associative) algebras onto some semisimple Banach algebras are automatically continuous.

scholarly journals Finite dimensionality, nilpotents and quasinilpotents in Banach algebras

1974 ◽  
Vol 19 (1) ◽  
pp. 45-49 ◽  
Author(s):  
J. Duncan ◽  
A. W. Tullo
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Finite Dimensional ◽  
Finite Dimensionality

In this note we present some rather loosely connected results on Banach algebras together with some illustrative examples. We consider various conditions on a Banach algebra which imply that it is finite dimensional. We also consider conditions which imply the existence of non-zero nilpotents, and hence the existence of finite dimensional subalgebras. In the setting of Banach algebras quasinilpotents figure more prominently than nilpotents. We give an example of a non-commutative Banach algebra in which 0 is the only quasinilpotent; this resolves a problem of Hirschfeld and Zelazko (4).

scholarly journals Character amenability of Banach algebras

2008 ◽  
Vol 144 (3) ◽  
pp. 697-706 ◽  
Author(s):  
MEHDI SANGANI MONFARED
Keyword(s):  
Banach Algebra ◽  
Group Algebra ◽  
Banach Algebras ◽  
Amenable Group ◽  
Fourier Algebra ◽  
Measure Algebra ◽  
Amenable Banach Algebra ◽  
Finite Dimensional ◽  
Character Amenability

AbstractWe introduce the notion of character amenable Banach algebras. We prove that character amenability for either of the group algebra L1(G) or the Fourier algebra A(G) is equivalent to the amenability of the underlying group G. Character amenability of the measure algebra M(G) is shown to be equivalent to G being a discrete amenable group. We also study functorial properties of character amenability. For a commutative character amenable Banach algebra A, we prove all cohomological groups with coefficients in finite-dimensional Banach A-bimodules, vanish. As a corollary we conclude that all finite-dimensional extensions of commutative character amenable Banach algebras split strongly.

scholarly journals The structure of homomorphisms from Banach algebras of differentiable functions into finite dimensional Banach algebras

1990 ◽  
Vol 13 (2) ◽  
pp. 387-392
Author(s):  
Viet Ngo
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Unit Interval ◽  
Differentiable Functions ◽  
Finite Dimensional

We show that the structure of continuous and discontinuous homomorphisms from the Banach algebraCn[0,1]ofntimes continuously differentiable functions on the unit interval[0,1]into finite dimensional Banach algebras is completely determined by higher point derivations.

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.

Spectral characterization of the socle in Jordan–Banach algebras

1995 ◽  
Vol 117 (3) ◽  
pp. 479-489 ◽  
Author(s):  
Bernard Aupetit
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Finite Rank ◽  
Banach Algebras ◽  
Spectral Characterization ◽  
Bounded Operators ◽  
Finite Rank Operators ◽  
Finite Dimensional ◽  
Minimal Idempotent

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

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