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Register a new userDerivations preserving quasinilpotent elements
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We consider a Banach algebra $A$ with the property that, roughly speaking, sufficiently many irreducible representations of $A$ on nontrivial Banach spaces do not vanish on all square zero elements. The class of Banach algebras with this property turns out to be quite large -- it includes $C^*$-algebras, group algebras on arbitrary locally compact groups, commutative algebras, $L(X)$ for any Banach space $X$, and various other examples. Our main result states that every derivation of $A$ that preserves the set of quasinilpotent elements has its range in the radical of $A$.
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We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of its support, and study these concepts from different sides : hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions as Regularity, $omega$-comparison, Corona Factorization Property, property R, etc.. are equivalent under mild assumptions.
Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $tau,$ let $S_0(M, tau)$ be the algebra of all $tau$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra $S_0(M, tau).$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $S_0(M, tau)$ is spatial.
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 $tau$ on $M$ let $S(M, tau)$ (resp. $S_0(M, tau)$) be the algebra of all $tau$-measurable (resp. $tau$-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 $S(M,tau)$) is inner, and each derivation on $S_0(M, tau)$ is spatial and implemented by an element from $S(M, tau).$
We study bounded operators defined in terms of the regular representations of the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary of this main orbit. As examples, we discuss various classes of magnetic pseudo-differential operators on nilpotent groups. We also prove localization and non-propagation properties associated to suitable parts of the essential spectrum. These are applied to twisted groupoids having a totally intransitive groupoid restriction at the boundary.
Let $q = e^{i theta} in mathbb{T}$ (where $theta in mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, i.e., $u$ and $v$ are unitaries such that $vu = quv$. In this paper we find the optimal constant $c = c_theta$ such that $u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are commuting unitaries. We show that [ c_theta = frac{4}{|u_theta+u_theta^*+v_theta+v_theta^*|}, ] where $u_theta, v_theta$ are the universal $q$-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating $q$-commuting unitaries to scalar multiples of $q$-commuting unitaries. The techniques that we develop allow us to give new and simple dilation theoretic proofs of well known results regarding the continuity of the field of rotations algebras. In particular, for the so-called Almost Mathieu Operator $h_theta = u_theta+u_theta^*+v_theta+v_theta^*$, we recover the fact that the norm $|h_theta|$ is a Lipshitz continuous function of $theta$, as well as the result that the spectrum $sigma(h_theta)$ is a $frac{1}{2}$-Holder continuous function in $theta$ with respect to the Hausdorff metric. In fact, we obtain this Holder continuity of the spectrum for every selfadjoint $*$-polynomial $p(u_theta,v_theta)$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.
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