Sum Of Commutators Research Articles

Sums of commutators in free probability

We study the linear span of commutators of free random variables and show that these are the only quadratic forms which satisfy the following equivalent properties:•preservation free infinite divisibility,•free and strong cancellation of odd cumulants,•symmetric distribution for any free family. The main combinatorial tool is an involution on non-crossing partitions.

2-local Lie *-automorphisms on factors

Let be a factor which is not of type . We prove that every surjective 2-local Lie *-automorphism of can be written in the form , where is an *-automorphism or the negative of an anti-*-automorphism, and h is a *-linear functional vanishing on all finite sums of commutators.

TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS

AbstractThe trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

Traces on pseudodifferential operators and sums of commutators

The aim of this paper is to show that various known characterizations of traces on classical pseudodifferential operators can actually be obtained by very elementary considerations on pseudodifferential operators, using only basic properties of these operators. Thereby, we give a unified treatment of the determinations of the space of traces (i) on ΨDOs of non-integer order or of regular parity-class, (ii) on integer order ΨDOs, (iii) on ΨDOs of non-positive orders in dimension ≥ 2, and (iv) on ΨDOs of non-positive orders in dimension 1.

Values of Noncommutative Polynomials, Lie Skew-Ideals and Tracial Nullstellensätze

A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of $d\times d$ matrices if and only if all of its values in the algebra of $d\times d$ matrices have zero trace.

Sums of commutators in ideals and modules of type II factors

Let ℳ be a factor of type II ∞ or II 1 having separable predual and let ℳ ¯ be the algebra of affiliated τ-measurable operators. We characterize the commutator space [ℐ,𝒥] for sub-(ℳ,ℳ)- bimodules ℐ and 𝒥 of ℳ ¯.

Principal Bimodules of Nest Algebras

We classify thewot-closed bimodules over a pair of nest algebras which are singly generated as algebraic and as norm-closed bimodules. The obstructions relate to the finite rank atoms. In particular, if both nest algebras have infinite multiplicity, then every wot-closed bimodule is (algebraically) principal. Another important special case is the ideal of strictly upper triangular operators, which is always principal; and the generator is a sum of commutators. In general, every countably generated wot-closed bimodule is singly generated, and we obtain explicit bounds on the number and norms of the terms in a factorization through the generator.

Expansion of an n-Point Function as a Sum of Commutators

We write the n-point function of currents as a sum over nested commutators, a form more suitable for certain current algebra calculations.