Displaying similar documents to “Derivations with a hereditary domain, II”

(Non-)amenability of ℬ(E)

Volker Runde (2010)

Banach Center Publications

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In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that ( E ) = ( E ) + i d E . Still, ℬ(ℓ²) is...

The kh-socle of a commutative semisimple Banach algebra

Youness Hadder (2020)

Mathematica Bohemica

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Let 𝒜 be a commutative complex semisimple Banach algebra. Denote by kh ( soc ( 𝒜 ) ) the kernel of the hull of the socle of 𝒜 . In this work we give some new characterizations of this ideal in terms of minimal idempotents in 𝒜 . This allows us to show that a “result” from Riesz theory in commutative Banach algebras is not true.

Δ -weak character amenability of certain Banach algebras

Hamid Sadeghi (2019)

Archivum Mathematicum

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In this paper we investigate Δ -weak character amenability of certain Banach algebras such as projective tensor product A ^ B and Lau product A × θ B , where A and B are two arbitrary Banach algebras and θ Δ ( B ) , the character space of B . We also investigate Δ -weak character amenability of abstract Segal algebras and module extension Banach algebras.

Approximate amenability for Banach sequence algebras

H. G. Dales, R. J. Loy, Y. Zhang (2006)

Studia Mathematica

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We consider when certain Banach sequence algebras A on the set ℕ are approximately amenable. Some general results are obtained, and we resolve the special cases where A = p for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras p ( ω ) .

Operator Connes-amenability of completely bounded multiplier Banach algebras

Bahman Hayati, Abasalt Bodaghi, Massoud Amini (2019)

Archivum Mathematicum

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For a completely contractive Banach algebra B , we find conditions under which the completely bounded multiplier algebra c b ( B ) is a dual Banach algebra and the operator amenability of B is equivalent to the operator Connes-amenability of c b ( B ) . We also show that, in this case, these are equivalent to the existence of a normal virtual operator diagonal.

C*-algebras have a quantitative version of Pełczyński's property (V)

Hana Krulišová (2017)

Czechoslovak Mathematical Journal

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A Banach space X has Pełczyński’s property (V) if for every Banach space Y every unconditionally converging operator T : X Y is weakly compact. H. Pfitzner proved that C * -algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that C ( K ) spaces for a compact Hausdorff space K enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

A theorem of Gel'fand-Mazur type

Hung Le Pham (2009)

Studia Mathematica

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Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection a α : α A there exist α ≠ β ∈ such that a α a β A is isomorphic to i = 1 r ( [ X ] / X d i [ X ] ) E , where d , . . . , d r , and E is either X [ X ] / X d [ X ] for some d₀ ∈ ℕ or a 1-dimensional i = 1 r [ X ] / X d i [ X ] -bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function f : [ 0 , 1 ] m X is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function f : [ 0 , 1 ] m X with respect to any norming subset there exists a separately increasing function g : [ 0 , 1 ] m such that the sets of...

Generalized derivations with power values on rings and Banach algebras

Abderrahman Hermas, Abdellah Mamouni, Lahcen Oukhtite (2024)

Mathematica Bohemica

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Let R be a prime ring and I a nonzero ideal of R . The purpose of this paper is to classify generalized derivations of R satisfying some algebraic identities with power values on I . More precisely, we consider two generalized derivations F and H of R satisfying one of the following identities: ...

Operator Figà-Talamanca-Herz algebras

Volker Runde (2003)

Studia Mathematica

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Let G be a locally compact group. We use the canonical operator space structure on the spaces L p ( G ) for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues O A p ( G ) of the classical Figà-Talamanca-Herz algebras A p ( G ) . If p ∈ (1,∞) is arbitrary, then A p ( G ) O A p ( G ) and the inclusion is a contraction; if p = 2, then OA₂(G) ≅ A(G) as Banach spaces, but not necessarily as operator spaces. We show that O A p ( G ) is a completely contractive Banach algebra for each p ∈ (1,∞), and that O A q ( G ) O A p ( G ) completely contractively...

Sequentially Right Banach spaces of order p

Mahdi Dehghani, Mohammad B. Dehghani, Mohammad S. Moshtaghioun (2020)

Commentationes Mathematicae Universitatis Carolinae

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We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order p , and those defined by the dual property, the sequentially Right * Banach spaces of order p for 1 p . These classes of Banach spaces are characterized by the notions of L p -limited sets in the corresponding dual space and R p * subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space X and a reflexive Banach...

On (Co)homology of triangular Banach algebras

Mohammad Sal Moslehian (2005)

Banach Center Publications

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Suppose that A and B are unital Banach algebras with units 1 A and 1 B , respectively, M is a unital Banach A,B-module, = A M 0 B is the triangular Banach algebra, X is a unital -bimodule, X A A = 1 A X 1 A , X B B = 1 B X 1 B , X A B = 1 A X 1 B and X B A = 1 B X 1 A . Applying two nice long exact sequences related to A, B, , X, X A A , X B B , X A B and X B A we establish some results on (co)homology of triangular Banach algebras.

Non-regularity for Banach function algebras

J. Feinstein, D. Somerset (2000)

Studia Mathematica

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Let A be a unital Banach function algebra with character space Φ A . For x Φ A , let M x and J x be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of J x is connected, and that if x does not belong to the Shilov boundary of A then the set y Φ A : M x J y has an infinite connected subset. Various related results are given.

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then embeds in L(X,Y), and ℓ¹ embeds complementably in X γ Y * . Applications to embeddings of c₀ in various spaces of operators are given.

On the existence of non-linear frames

Shah Jahan, Varinder Kumar, S.K. Kaushik (2017)

Archivum Mathematicum

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A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if 𝒳 is a Banach space such that 𝒳 * has a SRBF, then 𝒳 has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space 𝒳 has an approximative Schauder frame, then 𝒳 * has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity

Artur Michalak (2003)

Studia Mathematica

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) l i n f ( [ 0 , 1 ] ) ¯ contains an order isomorphic copy of D(0,1), (2) l i n f ( Q ) ¯ contains an isomorphic copy of C([0,1]), (3) l i n f ( [ 0 , 1 ] ) ¯ / l i n f ( Q ) ¯ contains an isomorphic copy of c₀(Γ) for some uncountable...

Generalized derivations on Lie ideals in prime rings

Basudeb Dhara, Sukhendu Kar, Sachhidananda Mondal (2015)

Czechoslovak Mathematical Journal

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Let R be a prime ring with its Utumi ring of quotients U and extended centroid C . Suppose that F is a generalized derivation of R and L is a noncentral Lie ideal of R such that F ( u ) [ F ( u ) , u ] n = 0 for all u L , where n 1 is a fixed integer. Then one of the following holds: ...

A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form F ( x ) : = F ̃ x ( 2 n + 1 ) where F ̃ : X 2 n + 1 Y is a continuous (2n+1)-linear operator.