It will be shown that for 1 < p < 2 the Schatten p-class is isometrically isomorphic to a subspace of the predual of a von Neumann algebra. Similar results hold for non-commutative Lp(N,τ)-spaces defined by a finite trace on a finite von Neumann algebra. The embeddings rely on a suitable notion of p-stable processes in the non-commutative setting.

Given a C ∗-algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [16], and show that A R is a C ∗-algebra, and τ extends to a semicontinuous semifinite trace on A R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A ′′ and can be approximated in measure by operators in A R, in analogy with improper Riemann integration. Unbounded Riemann measurable operators form a τ-a.e. bimodule on A R, denoted by AR, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on AR. As type II1 singular traces for a semifinite von Neumann algebra M with a normal semifinite faithful (non-atomic) trace τ have been defined as traces on M − M-bimodules of unbounded τ-measurable operators [5], type II1 singular traces for a C ∗-algebra A with a semicontinuous semifinite (non-atomic) trace τ are defined here as traces on A − A-bimodules of unbounded Riemann measurable operators (in AR) for any faithful representation of A. An application of singular traces for C ∗-algebras is contained in [6].

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A principal result of the paper is that if E is a symmetric Banach function space on the positive half-line with the Fatou property then, for all semi-finite von Neumann algebras (M; ), the absolute value mapping is Lipschitz continuous on the associated symmetric operator space E(M; ) with Lipschitz constant depending only on E if and only if E has non-trivial Boyd indices. It follows that if

. We prove that a C # -algebra A or a predual N# of a von Neumann algebra N has the Daugavet property if and only if A (or N) is non-atomic. We also prove a similar (although somewhat weaker) result for non-commutative Lp-spaces corresponding to non-atomic von Neumann algebras. 1. Introduction We say that a Banach space X has the Daugavet property (DP, in short) if every rank 1 operator T : X # X satisfies the Daugavet equation ||I X + T || = 1 + ||T ||, (1.1) where I X is the identity map on X . The study of the Daugavet property was inaugurated by I. Daugavet [4] in 1961; he proved that every compact operator on C([0, 1]) satisfies (1.1). Consequently, C([0, 1]) has the DP. Investigation of the Daugavet property continued in the seventies and eighties. In particular, J. Holub proved in [7] and [8] (see also [11]) that for any Hausdor# compact topological space K, C(K) has the DP if and only if K has no isolated points, and, similarly, L 1 () has the DP if and only if is n...

Let M be a von Neumann algebra with a faithful normal trace , and let H 1 be a finite, maximal, subdiagonal algebra of M. Fundamental theorems on conjugate functions for weak -Dirichlet algebras are shown to be valid for noncommutative H 1 . In particular the conjugation operator is shown to be a bounded linear map from L p (M; ) into L p (M; ) for 1 ! p ! 1, and to be a continuous map from L 1 (M; ) into L 1;1 (M; ). We also obtain that if an operator a is such that jaj log + jaj 2 L 1 (M; ) then its conjugate belongs to L 1 (M; ). Finally, we present some partial extensions of the classical Szego's theorem to the non-commutative setting.

Abstract. We define the notion of Connes-von Neumann spectral triple and consider the associated index problem. We compute the analytic Chern-Connes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for II∞ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue. Contents