On a conjecture of A. Bikchentaev

In \cite{bik1}, A. M. Bikchentaev conjectured that for positive $\tau-$measurable operators $a$ and $b$ affiliated with an arbitrary semifinite von Neumann algebra $\mathcal M$, the operator $b^{1/2}ab^{1/2}$ is submajorized by the operator $ab$ in the sense of Hardy-Littlewood. We prove this conjecture in full generality and present a number of applications to fully symmetric operator ideals, Golden-Thompson inequality and (singular) traces.Comment: "Spectral Analysis, Differential Equations and Mathematical Physics", H. Holden et al. (eds), Proceedings of Symposia in Pure Mathematics {\bf 87}, Amer. Math. Soc. (to appear

Measure Theory in Noncommutative Spaces

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 review 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

Commutator estimates in W∗W^*-algebras

Let $\mathcal{M}$ be a $W^*$-algebra and let $LS(\mathcal{M})$ be the algebra of all locally measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in LS(\mathcal{M})$ there exists a self-adjoint element $c_{_{0}}$ from the center of $LS(\mathcal{M})$, such that for any $\epsilon>0$ there exists a unitary element $u_\epsilon$ from $\mathcal{M}$, satisfying $|[a,u_\epsilon]| \geq (1-\epsilon)|a-c_{_{0}}|$. A corollary of this result is that for any derivation $\delta$ on $\mathcal{M}$ with the range in a (not necessarily norm-closed) ideal $I\subseteq\mathcal{M}$, the derivation $\delta$ is inner, that is $\delta(\cdot)=\delta_a(\cdot)=[a,\cdot]$, and $a\in I$. Similar results are also obtained for inner derivations on $LS(\mathcal{M})$.Comment: 30 page

Noncommutative Residues and a Characterisation of the Noncommutative Integral

We continue the study of the relationship between Dixmier traces and noncommutative residues initiated by A. Connes. The utility of the residue approach to Dixmier traces is shown by a characterisation of the noncommutative integral in Connes' noncommutative geometry (for a wide class of Dixmier traces) as a generalised limit of vector states associated to the eigenvectors of a compact operator (or an unbounded operator with compact resolvent), i.e. as a generalised quantum limit. Using the characterisation, a criteria involving the eigenvectors of a compact operator and the projections of a von Neumann subalgebra of bounded operators is given so that the noncommutative integral associated to the compact operator is normal, i.e. satisfies a monotone convergence theorem, for the von Neumann subalgebra.Comment: 15 page