Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

Let be a length function on a group G, and let M denote the operator of pointwise multiplication by on l2(G). Following Connes, M𝕃 can be used as a “Dirac” operator for the reduced group C*-algebra (G). It deûnes a Lipschitz seminorm on (G), which defines a metric on the state space of (G). We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-* topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on ûnitely generated nilpotent-by-finite groups.

Characterizations of bounded mean oscillation through commutators of bilinear singular integral operators

We characterize bounded mean oscillation in terms of the boundedness of commutators of various bilinear singular integral operators with pointwise multiplication. In particular, we study commutators of a wide class of bilinear operators of convolution type, including bilinear Calderón–Zygmund operators and the bilinear fractional integral operators.

Realizations of homogeneous Besov and Triebel–Lizorkin spaces and an application to pointwise multipliers

We study the dilation commuting realizations of the homogeneous Besov spaces [Formula: see text] or the homogeneous Triebel–Lizorkin spaces [Formula: see text] in the case p, q > 0, and either s - (n/p) ∉ ℕ0or s - (n/p) ∈ ℕ0and 0 < q ≤ 1 (0 < p ≤ 1 in the [Formula: see text]-case). We present an application to pointwise multiplication if s ≤ n/p.

Smooth pointwise multipliers of modulation spaces

Let 1 < p, q < ∞ and s, r ∈ ℝ. It is proved that any function in the amalgam space W(Hrp(ℝd), ℓ∞), where p' is the conjugate exponent to p and Hrp′ (ℝd) is the Bessel potential space, defines a bounded pointwise multiplication operator in the modulation space Msp,q(ℝd), whenever r > |s| + d

Bochner algebras and their compact multipliers

Let $\mathfrak{A}$ be a normed algebra with identity, Ω be a locally compact Hausdorf space and λ be a positive Radon measure on Ω with supp(λ) = Ω. In this paper, we establish a necessary and sufficient condition for L 1(Ω, $\mathfrak{A}$) to be an algebra with pointwise multiplication. Under this condition, we then characterize compact and weakly compact left multipliers on L 1(Ω, $\mathfrak{A}$).