Related papers: Banach limits and traces on $\mathcal L_{1,\infty}…
This paper introduces a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) on a general Marcinkiewicz space associated with an arbitrary semifinite von…
We develop general techniques and present an approach to solve the problem of constructing a maximal Banach ideal $({\frak A},{\bf A)}$ which does not satisfy a transfer of the norm estimation in the principle of local reflexivity to its…
We define a new class of singular positive traces on the ideal $\mathcal M_{1,\infty}$ of $B(H)$ generated by exponentiation invariant generalized limits. We prove that this new class is strictly contained in the class of all Dixmier…
We develop the theory of modulated operators in general principal ideals of compact operators. For Laplacian modulated operators we establish Connes' trace formula in its local Euclidean model and a global version thereof. It expresses…
The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1,1,...,1,...) is proved in a first course on functional…
In addition to Pisier's counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which {\it{are accessible}}. The first step is implied by the observation that a "good behaviour" of trace…
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…
We consider a continuous version of the classical notion of Banach limits, namely, positive linear functionals on $L^{\infty}(\mathbb{R}_+)$ invariant under translations $f(x) \mapsto f(x+s)$ of $L^{\infty}(\mathbb{R}_+)$ for every $s \ge…
This article - a part of a multipaper project investigating arithmetic mean ideals - investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., ``How many traces can an ideal support?"…
In the present paper we suggest a construction of symmetric functionals on a large class of symmetric spaces over a semifinite von Neumann algebra. This approach establishes a bijection between the symmetric functionals on symmetric spaces…
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…
Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We characterize compactness of $T$ in terms of differentiability of the Lipschitz functions defined on $X$ with values in another normed space $Z$. Furthermore, using a…
We prove perturbation results for traces on normed ideals in semifinite von Neumann algebra factors. This includes the case of Dixmier traces. In particular, we establish existence of spectral shift measures with initial operators being…
We introduce the notion of trace convexity for functions and respectively, for subsets of a compact topological space. This notion generalizes both classical convexity of vector spaces, as well as Choquet convexity for compact metric…
We prove that if A and B are bounded self-adjoint operators such that A-B belongs to the trace class, then |A| -|B| belongs to the principal ideal L_{1,\infty} in the algebra L(H) of all bounded operators on an infinite-dimensional Hilbert…
Among ideals of compact operators on a Hilbert space we identify a subclass of those closed with respect to the logarithmic submajorization. Within this subclass, we answer the questions asked by Pietsch \cite{Pietsch_nachrichten} and by…
Let $\mathcal H$ be an infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^*$-algebra of bounded (respectively, compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a…
In this paper we study the norm-attainment of positive operators between Banach lattices. By considering an absolute version of James boundaries, we prove that: If $E$ is a reflexive Banach lattice whose order is given by a basis and $F$ is…
Given a smooth manifold $M$ (with or without boundary), in this paper we study the regularisation of traces for the global pseudo-differential calculus in the context of non-harmonic analysis. Indeed, using the global pseudo-differential…
In this paper, we consider the space $\mathrm{BV}^{\mathbb A}(\Omega)$ of functions of bounded $\mathbb A$-variation. For a given first order linear homogeneous differential operator with constant coefficients $\mathbb A$, this is the space…