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Amenability for dual Banach algebras
Functional Analysis · Submitted 19 Mar 2002
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We confirm this impression.
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The abstract, sentence by sentence
We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closed submodule $A_\ast$ of $A^\ast$. The class of dual Banach algebras includes all $W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. We then develop two notions of amenability - Connes-amenability and strong Connes-amenability - which take the $w^\ast$-topology on dual Banach algebras into account. We relate the amenability of an Arens regular Banach algebra A to the (strong) Connes-amenability of $A^{\ast\ast}$; as an application, we show that there are reflexive Banach spaces with the approximation property such that L(E) is not Connes-amenable. We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudo-measures. Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones which avoids the equivalence of amenability and nuclearity for $C^\ast$-algebras.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closed submodule $A_\ast$ of $A^\ast$. | background 0.85 | 0.58 | 0.02 | asserted 1.00 |
| s1 | The class of dual Banach algebras includes all $W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. | background 1.00 | 0.09 | 0.03 | asserted 1.00 |
| s2 | The general impression is that amenable, dual Banach algebras are rather the exception than the rule. | background 0.97 | 0.06 | 0.15 | hedged possibility 0.71 |
| s3 | We confirm this impression. | result 0.71 | 0.58 | 0.27 | asserted 1.00 |
| s4 | We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. | result 0.72 | 0.94 | 0.15 | asserted 1.00 |
| s5 | We then develop two notions of amenability - Connes-amenability and strong Connes-amenability - which take the $w^\ast$-topology on dual Banach algebras into account. | contribution 0.92 | 0.95 | 0.05 | asserted 1.00 |
| s6 | We relate the amenability of an Arens regular Banach algebra A to the (strong) Connes-amenability of $A^{\ast\ast}$; as an application, we show that there are reflexive Banach spaces with the approximation property such that L(E) is not Connes-amenable. | result 0.60 | 0.93 | 0.10 | asserted 1.00 |
| s7 | We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudo-measures. | result 0.79 | 0.85 | 0.06 | asserted 1.00 |
| s8 | Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones which avoids the equivalence of amenability and nuclearity for $C^\ast$-algebras. | contribution 0.56 | 0.62 | 0.08 | asserted 1.00 |
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Original abstract
We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closed submodule $A_\ast$ of $A^\ast$. The class of dual Banach algebras includes all $W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. We then develop two notions of amenability - Connes-amenability and strong Connes-amenability - which take the $w^\ast$-topology on dual Banach algebras into account. We relate the amenability of an Arens regular Banach algebra A to the (strong) Connes-amenability of $A^{\ast\ast}$; as an application, we show that there are reflexive Banach spaces with the approximation property such that L(E) is not Connes-amenable. We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudo-measures. Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones which avoids the equivalence of amenability and nuclearity for $C^\ast$-algebras.
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