Amenability for dual Banach algebras

V. Runde. "Amenability for dual Banach algebras." Studia Mathematica 148.1 (2001): 47-66. <http://eudml.org/doc/284756>.

@article{V2001,
abstract = {We define a Banach algebra 𝔄 to be dual if 𝔄 = (𝔄⁎)* for a closed submodule 𝔄⁎ of 𝔄*. The class of dual Banach algebras includes all W*-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*-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**; 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*-algebras.},
author = {V. Runde},
journal = {Studia Mathematica},
keywords = {amenable dual Banach algebras; finite direct sums of full matrix algebras; strong Connes amenability; Arens regular Banach algebra; abstract harmonic analysis; amenable -algebras},
language = {eng},
number = {1},
pages = {47-66},
title = {Amenability for dual Banach algebras},
url = {http://eudml.org/doc/284756},
volume = {148},
year = {2001},
}

TY - JOUR
AU - V. Runde
TI - Amenability for dual Banach algebras
JO - Studia Mathematica
PY - 2001
VL - 148
IS - 1
SP - 47
EP - 66
AB - We define a Banach algebra 𝔄 to be dual if 𝔄 = (𝔄⁎)* for a closed submodule 𝔄⁎ of 𝔄*. The class of dual Banach algebras includes all W*-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*-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**; 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*-algebras.
LA - eng
KW - amenable dual Banach algebras; finite direct sums of full matrix algebras; strong Connes amenability; Arens regular Banach algebra; abstract harmonic analysis; amenable -algebras
UR - http://eudml.org/doc/284756
ER -