Let $S(0,1)$ be the $*$-algebra of all classes of Lebesgue measurable functions on the unit interval $(0,1)$ and let $(\mathcal{A},\left\|\cdot \right\|_\mathcal{A})$ be a complete symmetric $Δ$-normed $*$-subalgebra of $S(0,1)$, in which simple functions are dense, e.g., $L_\infty (0,1)$, $L_{\log}(0,1)$, $S(0,1)$ and the Arens algebra $L^ω(0,1)$ equipped with their natural $Δ$-norms. We show that there exists no non-trivial derivation $ δ: \mathcal{A} \to S(0,1)$ commuting with all dyadic translations of the unit interval. Let $\mathcal{M}$ be a type $II$ (or $I_\infty$) von Neumann algebra, $\mathcal{A}$ be its abelian von Neumann subalgebra, let $S(\mathcal{M})$ be the algebra of all measurable operators affiliated with $\mathcal{M}$.
We show that any non-trivial derivation $δ:\mathcal{A} \to S(\mathcal{A})$ can not be extended to a derivation on $S(\mathcal{M})$. In particular, we answer an untreated question in \cite{BKS1}.
Let $M$ be a type I von Neumann algebra with the center $Z,$ and let $LS(M)$ be the algebra of all locally measurable operators affiliated with $M.$ We prove that every $Z$-linear derivation on $LS(M)$ is inner. In particular all $Z$-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements from $LS(M).$
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 $\…
In the present paper derivations and *-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all L^0-linear derivations and L^{0}-linear *-automorphisms are inner. Moreover, it is proved that each L^0-linear automorphism of the algebra of all linear operators on a bo-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.
Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M.$ We give a complete description of all derivations on the algebra $L(M, τ).$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $L(M, τ)$ is inner.
Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M.$ We prove that if $A$ is a locally convex reflexive complete metrizable solid $\ast$-subalgebra in $L(M, τ),$ which can be embedded into a locally bounded weak Fréchet $M$-bimodule, then any derivation on $A$ is inner.
The paper is devoted to local derivations on the algebra $S(\mathcal{M},τ)$ of $τ$-measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ and a faithful normal semi-finite trace $τ.$ We prove that every local derivation on $S(\mathcal{M},τ)$ which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the condition of c…
Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $τ$ on $M$ let $S(M, τ)$ (resp. $S_0(M, τ)$) be the algebra of all $τ$-measurable (resp. $τ$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,τ)$) is inner, and each deriva…
Given a von Neumann algebra $M$ we introduce so called central extension $mix(M)$ of $M$. We show that $mix(M)$ is a *-subalgebra in the algebra $LS(M)$ of all locally measurable operators with respect to $M,$ and this algebra coincides with $LS(M)$ if and only if $M $ does not admit type II direct summands. We prove that if $M$ is a properly infinite von Neumann algebra then every additive derivation on the algebra $mix(M)$ is inner. This implies that on the algebra $LS(M)$, where $M$ is a type I$_\infty$ or a type III von Neumann algebra, all additive derivations are inner derivations.
This paper is devoted to local derivations on subalgebras on the algebra $S(M, τ)$ of all $τ$-measurable operators affiliated with a von Neumann algebra $M$ without abelian summands and with a faithful normal semi-finite trace $τ.$ We prove that if $\mathcal{A}$ is a solid $\ast$-subalgebra in $S(M, τ)$ such that $p\in \mathcal{A}$ for all projection $p\in M$ with finite trace, then every local derivation on the algebra $\mathcal{A}$ is a derivation. This result is new even in the case standard subalgebras on the algebra $B(H)$ of all bounded linear operators on a Hilbert space $H.$ We also ap…
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…
Let $M$ be a type I von Neumann algebra with the center $Z,$ a faithful normal semi-finite trace $τ.$ Let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M$ and let $S_0(M, τ)$ be the subalgebra in $L(M, τ)$ consisting of all operators $x$ such that given any $ε>0$ there is a projection $p\in\mathcal{P}(M)$ with $τ(p^{\perp})<\infty, xp\in M$ and $\|xp\|<ε.$ We prove that any $Z$-linear derivation of $S_0(M, τ)$ is spatial and generated by an element from $L(M, τ).$
We characterize those derivations from the convolution algebra $\ell^1({\mathbb Z}_+)$ to its dual which are weakly compact. In particular, we provide examples which are weakly compact but not compact. The characterization is combinatorial, in terms of "translation-finite" subsets of ${\mathbb Z}_+$, and we investigate how this notion relates to other notions of "smallness" for infinite subsets of ${\mathbb Z}_+$. In particular, we show that a set of strictly positive Banach density cannot be translation-finite; the proof has a Ramsey-theoretic flavour.
Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of these cases one can define a module of measurable 1-forms and a first-order exterior derivative. We give a general construction which applies to any metric space equipped with a sigma-finite measure and produces the desired result in all of the above cases. It also applies to an important class of Dirichlet spaces, where, however, the known first-order differe…
The aim of this article is to deepen the understanding of the derivation of $\mathrm{L}^p$-estimates of non-local operators. We review the $\mathrm{L}^p$-extrapolation theorem of Shen which builds on a real variable argument of Caffarelli and Peral and adapt this theorem to account for non-local weak reverse Hölder estimates. These non-local weak reverse Hölder estimates appear for example in the investigation of non-local elliptic integrodifferential operators. This originates from the fact that here only a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, and Sire. As an applic…
In this paper we study the properties of multiplication invariant (MI) operators acting on subspaces of the vector-valued space $L^2(X;\mathcal H)$. We characterize such operators in terms of range functions by showing that there is an isomorphism between the category of MI spaces (with MI operators as morphisms) and the category of measurable range functions whose morphisms are measurable range operators. We investigate how global properties of an MI operator are reflected by local pointwise properties of its corresponding range operator. We also establish several results about frames generat…
Let S be a bounded, Riemann measurable set in R^d, and L be a lattice. By a theorem of Fuglede, if S tiles R^d with translation set L, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles R^d with translation set L, S has a Riesz basis of exponentials. The proof is based on Meyer's quasicrystals.
Let $G$ be a locally compact group, and let $1\le p < \infty$. Consider the weighted $L^p$-space $L^p(G,ω)=\{f:\int|fω|^p<\infty\}$, where $ω:G\to \mathbb R$ is a positive measurable function. Under appropriate conditions on $ω$, $G$ acts on $L^p(G,ω)$ by translations. When is this action hypercyclic, that is, there is a function in this space such that the set of all its translations is dense in $L^p(G,ω)$? H. Salas (1995) gave a criterion of hypercyclicity in the case $G=\mathbb Z$ . Under mild assumptions, we present a corresponding characterization for a general locally compact group $G$. …
Let $L^1_\om(G)$ be a Beurling algebra on a locally compact abelian group $G$. We look for general conditions on the weight which allows the vanishing of continuous derivations of $L^1_\om(G)$. This leads us to introducing vector-valued Beurling algebras and considering the translation of operators on them. This is then used to connect the augmentation ideal to the behavior of derivation space. We apply these results to give examples of various classes of Beurling algebras which are weakly amenable, 2-weakly amenable or fail to be even 2-weakly amenable.
