Projections of uniqueness for L^p(G)

Carlo Cecchini and Alessandro Figà-Talamanca

Vol. 51 (1974), No. 1, 37–47

DOI: 10.2140/pjm.1974.51.37

Abstract

Let G be a locally compact, noncompact, unimodular group. For x ∈ G, we denote by L_x, the left translation operator defined on L²(G) by L_xf(y) = f(x⁻¹y). We let ℒ₂(G) be the closure, in the weak operator topology, of the algebra generated by the operators {L_x : x ∈ G}. For f ∈ L^p(G),1 ≦ p ≦ 2, we let L_f be the closed operator in L²(G), defined by L_fg = f ∗g, for g ∈ L¹(G) ∩L²(G). We prove, under a natural hypothesis on G, that for every 1 < p < 2, there exists a projection P ∈ℒ₂(G),P≠0, with the property that if f ∈ L^p(G), and PL_f = L_f, then f = 0. Thus P is a projection of uniqueness in the sense that the only element f ∈ L^p, such that the range of L_f is contained in the range of P is the zero element. Another way to express this result is the following: There exists a nontrivial closed subspace of L²(G), invariant under right translations and which contains no nonzero element of L^p(G).

Mathematical Subject Classification 2000

Primary: 43A15

Milestones

Received: 16 October 1972

Revised: 13 April 1973

Published: 1 March 1974