We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L (1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L (1,∞), i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L (1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L (1,∞) if and only if the sequence of singular numbers {sn(x)}n≥1 (in the descending order and counting the multiplicities) satisfies ‖x ‖ (1,∞):= 1 N supN≥1 Log(1+N) n=1 sn(x) < ∞. In this case, our characterization amounts to saying that a positive element x ∈ L (1,∞) is measurable if and only if limN→ ∞ 1 ∑N LogN n=1 sn(x) exists; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space L (1,∞) /L (1,∞) 0, where the space L (1,∞) 0 is the closure of all finite rank operators in L (1,∞) in the norm ‖. ‖ (1,∞).

Abstract. This paper has four main parts. In the first part we construct a noncommutative residue for hypoelliptic calculus on Heisenberg manifolds, that is, for the class of ΨHDO operators introdcued by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace induced on ΨHDOs of integer orders by the analytic extension of the usual trace to ΨHDOs of non-integer orders and it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the corresponding operator. In addition, we show that this noncommutative residue provides us with the unique trace up to constant multiple on the algebra of integer order ΨHDOs. In the second part, we give some analytic applications of this construction concerning zeta functions of hypoelliptic operators and logarithmic metric estimates for Green kernels of hypoelliptic operators, and we show that this noncommutative residue allows us to extend the Dixmier trace to the whole algebra of integer order ΨHDOs. In the third part, we present examples of computations of noncommutative residues for suitable powers of the horizontal sublaplacian and of Rumin’s contact Laplacian. In the fourth part, we present several applications in CR geometry. First, we give some examples of geometric computations of noncommutative residues of suitable powers of the horizontal sublaplacian and of the Kohn Laplacian. We then show that the logarithmic singularities of the Green kernels of the Gover-Graham are local CR invariants in the sense of Fefferman. Finally, we make use of framework of noncommutative geometry and of our noncommutative residue to define lower dimensional volumes in CR, e.g., we can give sense to the area of any 3-dimensional CR manifold. On the way we obtain a spectral interpretation of the Einstein-Hilbert action in CR geometry. 1.

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Abstract. To any spectral triple (A, D, H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that |D | −d has non trivial logarithmic Dixmier trace. Moreover, when d ∈ (0, ∞), there always exists a singular trace which is finite nonzero on |D | −d, giving rise to a noncommutative integration on A. Such results are applied to fractals in R, using Connes ’ spectral triple, and to limit fractals in R n, a class which generalises self-similar fractals, using a new spectral triple. The noncommutative dimension or measure can be computed in some cases. They are shown to coincide with the (classical) Hausdorff dimension and measure in the case of selfsimilar fractals. 1 Introduction. This paper is both a survey and an announcement of results concerning singular traces on B(H), and their application to the study of fractals in the framework of

2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9

Let X be a smooth manifold with boundary of dimension n ? 1. The operators of order \Gamman and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal L 1;1 (H) for the Hilbert space H = L 2 (X; E) \Phi L 2 (@X;F ) of L 2 sections in vector bundles E over X, F over @X. We show that, on these operators, Dixmier's trace can be computed in terms of the same expressions that determine the noncommutative residue. In particular it is independent of the averaging procedure. However, the noncommutative residue and Dixmier's trace are not multiples of each other unless the boundary is empty. As a corollary we show how to compute Dixmier's trace for parametrices or inverses of classical elliptic boundary value problems of the form Pu = f ; Tu = 0 with an elliptic differential operator P of order n in the interior and a trace operator T . In this particular situation, Dixmier's trace and the noncommutative residue do coincide up to a factor.

(Denmark) 2 and RFBR (5-01-00629)(Russia) 4. We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semi-finite von Neumann algebra. We find for p> 1 that the asymptotics of the zeta function determines an ideal strictly larger than L p, ∞ on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions. 1 1.

We consider the class of integral operators Q# on L (R+ ) of the form (Q# f)(x) = #(max{x, y})f(y)dy. We discuss necessary and su#cient conditions on # to insure that Q# is bounded, compact, or in the Schatten--von Neumann class S p , 1 < p < #.

After the introduction of spectral triples in Alain Connes ’ Noncommutative Geometry, singular traces on B(H) became a quite popular tool in operator algebras. With the aim of classifying them some papers have been written ([3],[8],[1], see also [4] for nonpositive traces), addressing in particular the question

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