Singular trace

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 2 (0,1), 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 self- similar fractals.

Abstract: The integral in noncommutative geometry (NCG) involves a non-standard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measure-theoretic view, however, the formulation contains several difficulties. We re- view results concerning the technical features of the integral in NCG and some outstanding problems in this area. The review is aimed for the general user of NCG.

Abstract: For a singular measure $\mu$, Ahlfors regular of order $\alpha>0,$ with compact support in $\mathbb{R}^{\mathbf{N}}$ and a pseudodifferential operator $\mathbf{A}$ of order $-l=-\mathbf{N}/2$ we consider the compact operator $\mathbf{T}(P,\mathbf{A}) = \mathbf{A}^*P\mathbf{A}.$ Here $P$ is the signed measure, $P=V\mu$ with density $V$ belonging to the Orlicz class $L^{\Psi,\mu}$ with $\Psi(t)=(t+1)\log(t+1)-t.$ Using eigenvalue estimates for such operators, obtained in \texttt{arXiv:2011.14877}, we establish eigenvalue asymptotics of $\mathbf{T}(P,\mathbf{A})$ for a class of measures, including the ones supported on uniformly rectifiable sets. These results lead to the measurability in the sense of A.Connes of operators $\mathbf{T}(P,\mathbf{A})$ and a formula for the singular trace of these operators, producing a noncommutative version of integral with respect to singular measure.

Abstract: For every symmetrically normed ideal $\mathcal{E}$ of compact operators, we give a criterion for the existence of a continuous singular trace on $\mathcal{E}$. We also give a criterion for the existence of a continuous singular trace on $\mathcal{E}$ which respects Hardy-Littlewood majorization. We prove that the class of all continuous singular traces on $\mathcal{E}$ is strictly wider than the class of continuous singular traces which respect Hardy-Littlewood majorization. We establish a canonical bijection between the set of all traces on $\mathcal{E}$ and the set of all symmetric functionals on the corresponding sequence ideal. Similar results are also proved in the setting of semifinite von Neumann algebras.