E_0-dilation of strongly commuting CP_0-semigroups

Orr Shalit

Home/Archive/arXiv

arXiv · 0707.1760

E_0-dilation of strongly commuting CP_0-semigroups

Orr Shalit

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that every strongly commuting pair of CP_0-semigroups has a minimal E_0-dilation. This is achieved in two major steps, interesting in themselves: 1: we show that a strongly commuting pair of CP_0-semigroups can be represented via a two parameter product system representation; 2: we prove that every fully coisometric product system representation has a fully coisometric, isometric dilation. In particular, we obtain that every commuting pair of CP_0-semigroups on B(H), H finite dimensional, has an E_0-dilation.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Orr Shalit. 2007-09-09. E_0-dilation of strongly commuting CP_0-semigroups. https://arxiv.org/abs/0707.1760

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Dynamical comparison for local homeomorphisms

We prove dynamical comparison for Deaconu--Renault groupoids associated to minimal surjective non-injective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension. As a corollary, the associated $C^*$-algebras are UCT Kirchberg algebras, recovering results by Carlsen--Thomsen via dynamical methods. In the zero-dimensional case, our result also verifies Matui's AH-conjecture for these groupoids using a recent breakthrough of Xin Li. Our proof combines techniques from both the purely infinite and stably finite regimes: We construct partial actions of non-abelian free groups as suitable ``large subgroupoids'' and establish comparison properties for these using the paradoxical towers technique developed by Gardella--Geffen--Kranz--Naryshkin. The boundary of the subgroupoid is controlled by a groupoid version of the topological small boundary property which we deduce from finite covering dimension of the unit space. As a byproduct, we prove the classical small boundary property for minimal actions of countable discrete groups on finite-dimensional compact metrizable spaces without any freeness assumption.

math.OA↗

arXiv2026

Maximal Algebraic Ideals in Nonunital $C^*$-Algebras

Motivated by Ozawa's question of whether every maximal algebraic two-sided ideal in a $C^*$-algebra must be closed, we study the existence of maximal algebraic two-sided ideals in nonunital $C^*$-algebras. We formulate singular-distribution estimates intrinsically in terms of lower semicontinuous $2$-quasitraces and develop a novel criterion, called a compact quasitracial comparison pair. These are used to control membership in algebraic ideals and to establish the nonexistence of maximal ideals in nonunital $C^*$-algebras under a closed finite-scale face condition. This nonexistence result applies to a wide class of simple $C^*$-algebras. Among others such as certain hereditary $C^*$-subalgebras of some stable algebras, certain $C^*$-algebras with purely infinite corona algebras, and certain crossed products, this class includes [i] stable simple $C^*$-algebras with an increasing projection approximate identity $(p_n)$ and a stably finite full corner having finite radius of comparison (e.g., all simple separable stable AF algebras); [ii] all nonunital, simple, separable, stably finite, $\mathcal Z$-stable $C^*$-algebras $A$ that have an approximate identity consisting of increasing projections $(p_n)$ and for which the simplex $\operatorname{QT}_2^1(A,p_1)$ of normalized traces at $p_1$ has finitely many extreme points. We also demonstrate a large class of $C^*$-algebras $E$, constructed from extensions of the $C^*$-algebras above, such that $E$ still has no maximal ideals. In particular, for these classes, Ozawa's question can be settled in an unexpected manner.

math.OA↗

arXiv2026

Quantum integrals (With an Appendix on the solution to Haagerup's weight problem)

We solve two long outstanding problems here. We first solve in ZFC a problem that the first author and Weaver posed ten years ago: a state on a von Neumann algebra is countably additive on orthogonal projections if and only if it is sequentially weak* continuous, equivalently, sequentially normal. Such states may be considered the quantum (countably additive) probability measures, or rather their extension to a noncommutative integral. The principal tools are known ideas from direct integral theory from the 1970s. Indeed we prove a disintegration theorem for states on products of sigma-finite von Neumann algebras. We give several applications. For example we give a new variant of Gleason theorem, extending countably additive probability measures on projections to sequentially weak* continuous states when the algebra has no type $I_2$ direct summand. We also establish normality and sequential normality criteria for weights, and give several other applications of our main result to weights. In forthcoming work we consider many applications to `quantum measure and integration theory', such as variants of Lebesgue's dominated convergence theorem for von Neumann algebras. The analytic tools and product-quotient construction developed here are used to prove in an Appendix by the first author that normality and sequential normality of arbitrary weights are determined by their restrictions to abelian von Neumann subalgebras, resolving Haagerup's Problem 1.11. (This may be moved to a companion paper.) That is, normality of weights is commutatively determined.

math.OA↗