Positive integrable elements relative to a left Hilbert algebra

Abstract

Let U be an achieved left Hilbert algebra. Let η∈D ♭ be an element such that π′(η) is a positive operator. Then, following M. A. Rieffel, η is called integrable if sup{( η| e) e∈ U and e e ♯ e 2} < + ∞. It is shown that η is integrable if and only if there is an element ζ∈ D flat; such π′(ζ) is positive and ζ is a square root of η in an appropriate sense. This is shown to be a generalization of Godement's well known theorem on the existence of a convolution square root for a continuous square-integrable positive-definite function on a locally compact group. An “integral” and an “ L 1 -norm” are then defined on the linear span of the positive integrable elements and the completion of this space, denoted by L 1( U ), is shown to be the predual of l ( U ). “Godement's theorem” is then used to investigate square-integrable representations of U .