Amenable dynamical systems over locally compact groups

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 $(G,X)$ .

Strongly Extreme Points and Approximation Properties

We show that if x is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at x, then x is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the suõcient conditions mentioned.

BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures

Let μ be a nonnegative Radon measure on ℝd that satisfies the growth condition that there exist constants C0 > 0 and n ∈ (0, d] such that for all x ∈ ℝd and r > 0, μ(B(x, r)) ≤ C0rn, where B(x, r) is the open ball centered at x and having radius r. In this paper, the authors prove that if f belongs to the BMO-type space RBMO(μ) of Tolsa, then the homogeneous maximal function S( f ) (when ℝd is not an initial cube) and the inhomogeneous maximal function ℳS( f ) (when ℝd is an initial cube) associated with a given approximation of the identity S of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, S and ℳS are bounded from RBMO(μ) to the BLO-type space RBLO(μ). The authors also prove that the inhomogeneous maximal operator ℳS is bounded from the local BMO-type space rbmo(μ) to the local BLO-type space rblo(μ).

On commuting approximation properties of Banach spaces

Let a, c ≥ 0 and let B be a compact set of scalars. We show that if X is a Banach space such that the canonical projection π from X*** onto X* satisfies the inequalityand 1 ≤ λ < max |B| + c, then every λ-commuting bounded compact approximation of the identity of X is shrinking. This generalizes a theorem by Godefroy and Saphar from 1988. As an application, we show that under the conditions described above both X and X* have the metric compact approximation property (MCAP). Relying on the Willis construction, we show that the commuting MCAP does not imply the approximation property.

A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces

We work on RD-spaces𝒳, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in𝒳. An important example is the Carnot-Carathéodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spacesHp(𝒳)and local Hardy spaceshp(𝒳)on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces whenp≥1. The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces, inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, and BMO are also presented. Moreover, we prove boundedness results on these Besov and Triebel-Lizorkin spaces for classes of singular integral operators, which include non-isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn-Laplacian on certain classes of model domains inℂN. Our theory applies in a wide range of settings.