Geodesic
Averaging operators in non commutative Lp spaces I
Glasgow mathematical journal, Tome 24 (1983) no. 1, pp. 71-74

Voir la notice de l'article provenant de la source Cambridge University Press

The origin of the theory of averaging operators is explained in [1]. The theory has been developed on spaces of continuous functions that vanish at infinity by Kelley in [3] and on the Lp spaces of measure theory by Rota [5]. The motivation for this paper arose out of the latter paper. The aim of this paper is to prove a generalisation of Rota's main representation theorem (every average is a conditional expectation) in the context of a ‘non commutative integration’. This context is as follows. Let be a finite von Neumann algebra and φ a faithful normal finite trace on such that φ(I) = 1, where I is the identity of . We can construct the Banach spaces Lp (, φ), where 1 ≤ p < °, with norm ∥x∥p = φ(÷x÷p)1/p, of possibly unbounded operators affiliated with , as in [9]. We note that is dense in Lp(, φ). These spaces share many of the features of the Lp spaces of measure theory; indeed if is abelian then Lp(,φ) is isometrically isomorphic to Lp of some measure space. 
Barnett, Christopher. Averaging operators in non commutative Lp spaces I. Glasgow mathematical journal, Tome 24 (1983) no. 1, pp. 71-74. doi: 10.1017/S0017089500005073
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