Abstract
We establish several new characterizations of amenable
$W^*$
- and
$C^*$
-dynamical systems over arbitrary locally compact groups. In the
$W^*$
-setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of
$(M,G,\alpha )$
converging point weak* to the identity of
$G\bar {\ltimes }M$
. In the
$C^*$
-setting, we prove that amenability of
$(A,G,\alpha )$
is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product
$G\ltimes A$
, as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted
$C^*$
-dynamical systems, Hilbert
$C^*$
-modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When
$Z(A^{**})=Z(A)^{**}$
, it follows that amenability is equivalent to the1-positive approximation property of Exel and Ng [Approximation property of
$C^*$
-algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when
$A=C_0(X)$
is commutative, amenability of
$(C_0(X),G,\alpha )$
coincides with topological amenability of the G-space
$(G,X)$
.
The approximation property and exactness of locally compact groups
