Vol. 51, No. 1, 1974

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Projections of uniqueness for Lp(G)

Carlo Cecchini and Alessandro Figà-Talamanca

Vol. 51 (1974), No. 1, 37–47

Let G be a locally compact, noncompact, unimodular group. For x G, we denote by Lx, the left translation operator defined on L2(G) by Lxf(y) = f(x1y). We let 2(G) be the closure, in the weak operator topology, of the algebra generated by the operators {Lx : x G}. For f Lp(G),1 p 2, we let Lf be the closed operator in L2(G), defined by Lfg = 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),P0, with the property that if f Lp(G), and PLf = Lf, then f = 0. Thus P is a projection of uniqueness in the sense that the only element f Lp, such that the range of Lf 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 Lp(G).

Mathematical Subject Classification 2000
Primary: 43A15
Received: 16 October 1972
Revised: 13 April 1973
Published: 1 March 1974
Carlo Cecchini
Alessandro Figà-Talamanca