applied when the von Neumann algebra M is type I or type II. When M = B(H) acting on the Hilbert space L2(H) of Hilbert–Schmidt operators, it gives the well-known theory of non-commutative Lp spaces Lp (H) due to Schatten and von Neumann. When M is in addition a finite von Neumann algebra, for example a type II1 factor, then every affiliated operator is automatically measurable, so the affiliated operators form a *-algebra, as originally observed in the first paper of Murray and von Neumann. In this case M is a von Neumann regular ring: for on the closure of its image |A| has a measurable inverse B and then T = BV* defines a measurable operator with ATA = A. Of course in the classical case when X is a probability space and M = L∞ (X), we simply recover the *-algebra of measurable functions on X. If however M is type III, the theory takes a quite different form. Indeed in this case, thanks to the Tomita–Takesaki theory, it is known that the non-commutative Lp spaces are no longer realised by operators affiliated with the von Neumann algebra. As Connes showed, these spaces can be realised as unbounded operators only by using a certain positive power of the reference modular operator. Instead of being characterised by the simple affiliation relation UAU* = A, there is a more complicated bimodule relation involving the analytic continuation of the modular automorphism group. References A. Connes, Non-commutative geometry, J. Dixmier, Von Neumann algebras, [Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars (1957 & 1969)] W. Lück, L2-Invariants: Theory and Applications to Geometry and K-Theory, (Chapter 8: the algebra of affiliated operators) F. J. Murray and J. von Neumann, Rings of Operators, Annals of Mathematics 37 (1936), 116–229 (Chapter XVI). E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103–116. M. Takesaki, Theory of Operator Algebras I, II, III'', Operator theory Von Neumann algebras