Harmonic operators: The dual perspective

The study of harmonic functions on a locally compact group G has recently been transferred to a "non-commutative" setting in two different directions: Chu and Lau replaced the algebra L ∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L ∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand Jaworski and the first author replaced L ∞(G) by B (L2(G)) to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action of σ on VN(G) can be extended to B (L2(G)). We study the corresponding space ℋ σ̄ "σ-harmonic operators" i.e. fixed points in B (L2(G)) under the action of σ. We show under mild conditions on either σ or G that is in fact a von Neumann subalgebra of B (L2(G)). Our investigation of ℋσ̄ relies in particular on a notion of support for an arbitrary operator in B (L 2(G)) that extends Eymard's definition for elements of VN(G). Finally we present an approach to ℋσ̄ via ideals in T (L2(G)) where T (L2(G)) denotes the trace class operators on L 2(G) but equipped with a product different from composition as it was pioneered for harmonic functions by Willis

Column and row operator spaces over QSLp-spaces and their use in abstract harmonic analysis

The notions of column and row operator space were extended by A. Lambert from Hilbert spaces to general Banach spaces. In this paper, we use column and row spaces over quotients of subspaces of general Lp-spaces to equip several Banach algebras occurring naturally in abstract harmonic analysis with canonical, yet not obvious operator space structures that turn them into completely bounded Banach algebras. We use these operator space structures to gain new insights on those algebras

Operator space structure and amenability for Figà-Talamanca-Herz algebras

Column and row operator spaces - which we denote by COL and ROW, respectively - over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p,p′ ∈ (1, ∞) with 1/p + 1/p′ = 1, we use the operator space structure on CB(COL(Lp′ (G))) to equip the Figà-Talamanca-Herz algebra Ap(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p≤q≤2 or 2≤q≤p and amenable G, the canonical inclusion Aq (G) ⊂ Ap (G) is completely bounded (with cb-norm at most KG 2, where KG, is Grothendieck's constant). As an application, we show that G is amenable if and only if Ap(G) is operator amenable for all - and equivalently for one - p ∈ (1, ∞); this extends a theorem by Ruan