The Grothendieck property in Marcinkiewicz spaces
The main pupose of this paper is to fully characterize continuous concave functions $ψ$ such that the corresponding Marcinkiewicz Banach function space $M_{ψ}$ is a Grothendieck space.
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arXiv subjects
F. A. Sukochev
Publications and source records attributed to F. A. Sukochev.
Innerness of continuous derivations on algebras of locally measurable operators
It is established that every derivation continuous with respect to the local measure topology acting on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ is necessary inner. If $\mathcal{M}$ is a properly infinite von Neumann algebra, then every derivation on $LS(\mathcal{M})$ is inner. In addition, it is proved that any derivation on $\mathcal{M}$ with values in Banach $\mathcal{M}$-bimodule of locally measurable operators is inner.
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Continuity of derivations in algebras of locally measurable operators
We prove that any derivation of the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a properly infinite von Neumann algebra $\mathcal{M}$ is continuous with respect to the local measure topology $t(\mathcal{M})$. Building an extension of a derivation $δ:\mathcal{M}\longrightarrow LS(\mathcal{M})$ up to a derivation from $LS(\mathcal{M})$ into $LS(\mathcal{M})$, it is further established that any derivation from $\mathcal{M}$ into $LS(\mathcal{M})$ is $t(\mathcal{M})$-continuous.
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Derivations on symmetric quasi-Banach ideals of compact operators
Let $\mathcal{I,J}$ be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space $H$, let $\mathcal{J:I}$ be a space of multipliers from $\mathcal{I}$ to $\mathcal{J}$. Obviously, ideals $\mathcal{I}$ and $\mathcal{J}$ are quasi-Banach algebras and it is clear that ideal $\mathcal{J}$ is a bimodule for $\mathcal{I}$. We study the set of all derivations from $\mathcal{I}$ into $\mathcal{J}$. We show that any such derivation is automatically continuous and there exists an operator $a\in\mathcal{J:I}$ such that $δ(\cdot)=[a,\cdot]$, moreover $\|a\|_{\mathcal{B}(H)}\leq\|δ\|_\mathcal{I\to J}\leq 2C\|a\|_\mathcal{J:I}$, where $C$ is the modulus of concavity of the quasi-norm $\|\cdot\|_\mathcal{J}$. In the special case, when $\mathcal{I=J=K}(H)$ is a symmetric Banach ideal of compact operators on $H$ our result yields the classical fact that any derivation $δ$ on $\mathcal{K}(H)$ may be written as $δ(\cdot)=[a,\cdot]$, where $a$ is some bounded operator on $H$ and $\|a\|_{\mathcal{B}(H)}\leq\|δ\|_\mathcal{I\to I}\leq 2\|a\|_{\mathcal{B}(H)}$.
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Derivations in the Banach ideals of $τ$-compact operators
Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semi-finite trace $τ$ and let $S_0(τ)$ be the algebra of all $τ$-compact operators affiliated with $\mathcal{M}$. Let $E(τ)\subseteq S_0(τ)$ be a symmetric operator space (on $\mathcal{M}$) and let $\mathcal{E}$ be a symmetrically-normed Banach ideal of $τ$-compact operators in $\mathcal{M}$. We study (i) derivations $δ$ on $\mathcal{M}$ with the range in $E(τ)$ and (ii) derivations on the Banach algebra $\mathcal{E}$. In the first case our main results assert that such derivations are continuous (with respect to the norm topologies) and also inner (under some mild assumptions on $E(τ)$). In the second case we show that any such derivation is necessarily inner when $\mathcal{M}$ is a type $I$ factor. As an interesting application of our results for the case (i) we deduce that any derivation from $\mathcal{M}$ into an $L_p$-space, $L_p(\mathcal{M},τ)$, ($1<p<\infty$) associated with $\mathcal{M}$ is inner.
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Best constants in Rosenthal-type inequalities and the Kruglov operator
Let $X$ be a symmetric Banach function space on $[0,1]$ with the Kruglov property, and let $\mathbf{f}=\{f_k\}_{k=1}^n$, $n\ge1$ be an arbitrary sequence of independent random variables in $X$. This paper presents sharp estimates in the deterministic characterization of the quantities \[\Biggl\|\sum_{k=1}^nf_k\Biggr\|_X,\Biggl\|\Biggl(\sum_{k=1}^n|f_k|^p\Biggr)^{1/p}\Biggr\|_X,\qquad 1\leq p<\infty,\] in terms of the sum of disjoint copies of individual terms of $\mathbf{f}$. Our method is novel and based on the important recent advances in the study of the Kruglov property through an operator approach made earlier by the authors. In particular, we discover that the sharp constants in the characterization above are equivalent to the norm of the Kruglov operator in $X$.
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$ζ-$function and heat kernel formulae
We present a systematic study of asymptotic behavior of (generalised) $ζ-$functions and heat kernels used in noncommutative geometry and clarify their connections with Dixmier traces. We strengthen and complete a number of results from the recent literature and answer (in the affirmative) the question raised by M. Benameur and T. Fack \cite{BF}.
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Commutator estimates in $W^*$-factors
Let $\mathcal{M}$ be a $W^*$-factor and let $S\left( \mathcal{M} \right) $ be the space of all measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in S(\mathcal{M})$ there exists a scalar $λ_0\in\mathbb{R}$, such that for all $\varepsilon > 0$, there exists a unitary element $u_\varepsilon$ from $\mathcal{M}$, satisfying $|[a,u_\varepsilon]| \geq (1-\varepsilon)|a-λ_0\mathbf{1}|$. A corollary of this result is that for any derivation $δ$ on $\mathcal{M}$ with the range in an ideal $I\subseteq\mathcal{M}$, the derivation $δ$ is inner, that is $δ(\cdot)=δ_a(\cdot)=[a,\cdot]$, and $a\in I$. Similar results are also obtained for inner derivations on $S(\mathcal{M})$.
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Kruglov operator and operators defined by random permutations
The Kruglov property and the Kruglov operator play an important role in the study of geometric properties of r.i. function spaces. We prove that the boundedness of the Kruglov operator in a r.i. space is equivalent to the uniform boundedness on this space of a sequence of operators defined by random permutations. It is shown also that there is no minimal r.i. space with the Kruglov property.
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Lidskii-type formulae for Dixmier traces
We establish several analogues of the classical Lidskii Theorem for some special classes of singular traces (Dixmier traces and Connes-Dixmier traces) used in noncommutative geometry.
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Orbits in symmetric spaces, II
Suppose $E$ is fully symmetric Banach function space on $(0,1)$ or $(0,\infty)$ or a fully symmetric Banach sequence space. We give necessary and sufficient conditions on $f\in E$ so that its orbit $Ω(f)$ is the closed convex hull of its extreme points. We also give an application to symmetrically normed ideals of compact operators on a Hilbert space.
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Some remarks on derivations in algebras of measurable operators
This paper is concerned with derivations in algebras of (unbounded) operators affiliated with a von Neumann algebra $\mathcal{M}$. Let $\mathcal{% A}$ be one of the algebras of measurable operators, locally measurable operators or, $τ$-measurable operators. We present a complete description of von Neumann algebras $\mathcal{M}$ of type $I$ in terms of their central projections such that every derivation in $\mathcal{A}$ is inner. It is also shown that every derivation in the algebra $LS(\mathcal{M})$ of all locally measurable operators with respect to a properly infinite von Neumann algebra $\mathcal{M}$ vanishes on the center of $LS(\mathcal{M})$.
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Derivations in algebras of operator-valued functions
In this paper we study derivations in subalgebras of $L_{0}^{wo}(ν;% \mathcal{L}(X)) $, the algebra of all weak operator measurable funtions $f:S\to \mathcal{L}(X) $, where $% \mathcal{L}(X) $ is the Banach algebra of all bounded linear operators on a Banach space $X$. It is shown, in particular, that all derivations on $L_{0}^{wo}(ν;\mathcal{L}(X)) $ are inner whenever $X$ is separable and infinite dimensional. This contrasts strongly with the fact that $L_{0}^{wo}(ν;\mathcal{L}(X)) $ admits non-trivial non-inner derivations whenever $X$ is finite dimensional and the measure $ν$ is non-atomic. As an application of our approach, we study derivations in various algebras of measurable operators affiliated with von Neumann algebras.
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The spectral shift function and spectral flow
This paper extends Krein's spectral shift function theory to the setting of semifinite spectral triples. We define the spectral shift function under these hypotheses via Birman-Solomyak spectral averaging formula and show that it computes spectral flow.
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Operator integrals, spectral shift and spectral flow
We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.
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The Dixmier trace and asymptotics of zeta functions
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 {\mathcal L}^{p,\infty} on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions.
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Dixmier traces and some applications to noncommutative geometry
This is a survey of some recent advances in the theory of singular traces in which the authors have played some part and which were inspired by questions raised by the book of Alain Connes (Noncommutative Geometry, Academic Press 1994). There are some original proofs and ideas but most of the results have appeared elsewhere. Detailed information on the contents is contained in the Introduction.
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An Analytic Approach to Spectral Flow in von Neumann Algebras
The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by Breuer-Fredholm operators in a semifinite von Neumann algebra. The first part of this paper gives a brief account of this theory extending and refining earlier results. It is then applied in the latter parts of the paper to a series of examples. One of the most powerful tools is an integral formula for spectral flow. This integral formula was known for Dirac operators in a variety of forms ever since the fundamental papers of Atiyah, Patodi and Singer. One of the purposes of this exposition is to make contact with this early work so that one can understand the recent developments in a proper historical context. In addition we show how to derive these spectral flow formulae in the setting of Dirac operators on (non-compact) covering spaces of a compact spin manifold using the adiabatic method. Finally we relate our work to that of Coburn, Douglas, Schaeffer and Singer on Toeplitz operators with almost periodic symbol. We generalise their work to cover the case of matrix valued almost periodic symbols on \R^N using some ideas of Shubin.
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