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1
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Range inclusion results for derivations on noncommutative Banach algebras

100%
Runde V.
Studia Mathematica
|
1993
|
tom 105
|
nr 2
159-172
EN
Let A be a Banach algebra, and let D : A → A be a (possibly unbounded) derivation. We are interested in two problems concerning the range of D: 1. When does D map into the (Jacobson) radical of A? 2. If [a,Da] = 0 for some a ∈ A, is Da necessarily quasinilpotent? We prove that derivations satisfying certain polynomial identities map into the radical. As an application, we show that if [a,[a,[a,Da]]] lies in the prime radical of A for all a ∈ A, then D maps into the radical. This generalizes a result by M. Mathieu and the author which asserts that every centralizing derivation on a Banach algebra maps into the radical. As far as the second question is concerned, we are unable to settle it, but we obtain a reduction of the problem and can prove the quasinilpotency of Da under commutativity assumptions slightly stronger than [a,Da] = 0.
2
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(Non-)amenability of ℬ(E)

100%
Runde V.
Banach Center Publications
|
2010
|
tom 91
|
nr 1
339-351
EN
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 $ℬ(E) = 𝒦(E) + ℂid_{E}$. Still, ℬ(ℓ²) is not amenable, and in the past decade, $ℬ(ℓ^{p})$ was found to be non-amenable for p = 1,2,∞ thanks to the work of C. J. Read, G. Pisier, and N. Ozawa. We survey those results, and then-based on joint work with M. Daws-outline a proof that establishes the non-amenability of $ℬ(ℓ^{p})$ for all p ∈ [1,∞].
3
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When is there a discontinuous homomorphism from L¹(G)?

100%
Runde V.
Studia Mathematica
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1994
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tom 110
|
nr 1
97-104
EN
Let A be an A*-algebra with enveloping C*-algebra C*(A). We show that, under certain conditions, a homomorphism from C*(A) into a Banach algebra is continuous if and only if its restriction to A is continuous. We apply this result to the question in the title.
4
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Operator Figà-Talamanca-Herz algebras

100%
Runde V.
Studia Mathematica
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2003
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tom 155
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nr 2
153-170
EN
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 $OA_{p}(G)$ of the classical Figà-Talamanca-Herz algebras $A_{p}(G)$. If p ∈ (1,∞) is arbitrary, then $A_{p}(G) ⊂ OA_{p}(G)$ 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 $OA_{p}(G)$ is a completely contractive Banach algebra for each p ∈ (1,∞), and that $OA_{q}(G) ⊂ OA_{p}(G)$ completely contractively for amenable G if 1 < p ≤ q ≤ 2 or 2 ≤ q ≤ p < ∞. Finally, we characterize the amenability of G through the existence of a bounded approximate identity in $OA_{p}(G)$ for one (or equivalently for all) p ∈ (1,∞).
5
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Erratum to "Can $ℬ(ℓ^{p})$ ever be amenable?" (Studia Math. 188 (2008), 151-174)

64%
Daws M., Runde V.
Studia Mathematica
|
2009
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tom 195
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nr 3
297-298
6
Content available remote

Can $ℬ(ℓ^{p})$ ever be amenable?

64%
Daws M., Runde V.
Studia Mathematica
|
2008
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tom 188
|
nr 2
151-174
EN
It is known that $ℬ(ℓ^{p})$ is not amenable for p = 1,2,∞, but whether or not $ℬ(ℓ^{p})$ is amenable for p ∈ (1,∞) ∖ {2} is an open problem. We show that, if $ℬ(ℓ^{p})$ is amenable for p ∈ (1,∞), then so are $ℓ^{∞}(ℬ(ℓ^{p}))$ and $ℓ^{∞}(𝓚(ℓ^{p}))$. Moreover, if $ℓ^{∞}(𝓚(ℓ^{p}))$ is amenable so is $ℓ^{∞}(𝕀,𝓚(E))$ for any index set 𝕀 and for any infinite-dimensional $ℒ^{p}$-space~E; in particular, if $ℓ^{∞}(𝓚(ℓ^{p}))$ is amenable for p ∈ (1,∞), then so is $ℓ^{∞}(𝓚(ℓ^{p} ⊕ ℓ²))$. We show that $ℓ^{∞}(𝓚(ℓ^{p} ⊕ ℓ²))$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter 𝒰 over ℕ, we exhibit a closed left ideal of $(𝓚(ℓ^{p}))_{𝒰}$ lacking a right approximate identity, but enjoying a certain very weak complementation property.
7
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Beurling-Figà-Talamanca-Herz algebras

51%
Öztop S., Runde V., Spronk N.
Studia Mathematica
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2012
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tom 210
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nr 2
117-135
EN
For a locally compact group G and p ∈ (1,∞), we define and study the Beurling-Figà-Talamanca-Herz algebras $A_{p}(G,ω)$. For p = 2 and abelian G, these are precisely the Beurling algebras on the dual group Ĝ. For p = 2 and compact G, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We prove that a locally compact group G is amenable if and only if one-and, equivalently, every-Beurling-Figà-Talamanca-Herz algebra $A_{p}(G,ω)$ has a bounded approximate identity.
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