arXiv subjects

Christopher Jankowski

Publications and source records attributed to Christopher Jankowski.

7 recordsLinked to original sources

arXiv2018

Classification of q-pure q-weight maps over finite dimensional Hilbert spaces

An $E_0$-semigroup of $B(H)$ is a one parameter strongly continuous semigroup of $*$-endomorphisms of $B(H)$ that preserve the identity. Every $E_0$-semigroup that possesses a strongly continuous intertwining semigroup of isometries is cocycle conjugate to an $E_0$-semigroup induced by the Bhat induction of a $CP$-flow over a separable Hilbert space $K$. We say an $E_0$-semigroup $α$ is $q$-pure if the $CP$-subordinates $β$ of norm one (i.e. $\Vertβ_t(I)\Vert = 1$ and $α_t-β_t$ is completely positive for all $t \geq 0$) are totally ordered in the sense that if $β$ and $γ$ are two $CP$-subordinates of $α$ of norm one, then $β\geq γ$ or $γ\geq β$. This paper shows how to construct and classify all $q$-pure $E_0$-semigroups induced by $CP$-flows over a finite-dimensional Hilbert space $K$ up to cocycle conjugacy.

math.OA↗

arXiv2011

Aligned CP-semigroups

A CP-semigroup is aligned if its set of trivially maximal subordinates is totally ordered by subordination. We prove that aligned spatial E_0-semigroups are prime: they have no non-trivial tensor product decompositions up to cocycle conjugacy. As a consequence, we establish the existence of uncountably many non-cocycle conjugate E_0-semigroups of type II_0 which are prime.

math.OA↗

arXiv2011

E_0-semigroups and q-purity

An E_0-semigroup is called q-pure if it is a CP-flow and its set of flow subordinates is totally ordered by subordination. The range rank of a positive boundary weight map is the dimension of the range of its dual map. Let K be a separable Hilbert space. We describe all q-pure E_0-semigroups of type II_0 which arise from boundary weight maps with range rank one over K. We also prove that no q-pure E_0-semigroups of type II_0 arise from boundary weight maps with range rank two over K. In the case when K is finite-dimensional, we provide a criterion to determine if two boundary weight maps of range rank one over K give rise to cocycle conjugate q-pure E_0-semigroups.

math.OA↗

arXiv2011

Gauge groups of E_0-semigroups obtained from Powers weights

The gauge group is computed explicitly for a family of E_0-semigroups of type II_0 arising from the boundary weight double construction introduced earlier by Jankowski. This family contains many E_0-semigroups which are not cocycle cocycle conjugate to any examples whose gauge groups have been computed earlier. Further results are obtained regarding the classification up to cocycle conjugacy and up to conjugacy for boundary weight doubles $(ϕ, ν)$ in two separate cases: first in the case when $ϕ$ is unital, invertible and q-pure and $ν$ is any type II Powers weight, and secondly when $ϕ$ is a unital $q$-positive map whose range has dimension one and $ν(A) = (f, Af)$ for some function f such that $(1-e^{-x})^{1/2}f(x) \in L^2(0,\infty)$. All E_0-semigroups in the former case are cocycle conjugate to the one arising simply from $ν$, and any two E_0-semigroups in the latter case are cocycle conjugate if and only if they are conjugate.

math.OA↗

arXiv2010

A family of non-cocycle conjugate E_0-semigroups obtained from boundary weight doubles

We have seen that if ϕ: M_n(\C) \rightarrow M_n(\C) is a unital q-positive map and νis a type II Powers weight, then the boundary weight double (ϕ, ν) induces a unique (up to conjugacy) type II_0 E_0-semigroup. Let ϕ: M_n(\C) \rightarrow M_n(\C) and ψ: M_{n'}(\C) \rightarrow M_{n'}(\C) be unital rank one q-positive maps, so for some states ρ\in M_n(\C)^* and ρ' \in M_{n'}(\C)^*, we have ϕ(A)=ρ(A)I_n and ψ(D) = ρ'(D)I_{n'} for all A \in M_n(\C) and D \in M_{n'}(\C). We find that if νand ηare arbitrary type II Powers weights, then (ϕ, ν) and (ψ, η) induce non-cocycle conjugate E_0-semigroups if ρand ρ' have different eigenvalue lists. We then completely classify the q-corners and hyper maximal q-corners from ϕto ψ, obtaining the following result: If νis a type II Powers weight of the form ν(\sqrt{I - Λ(1)} B \sqrt{I - Λ(1)})=(f,Bf), then the E_0-semigroups induced by (ϕ,ν) and (ψ, ν) are cocycle conjugate if and only if n=n' and ϕand ψare conjugate.

math.OA↗

arXiv2010

On type II_0 E_0-semigroups induced by boundary weight doubles

Powers has shown that each spatial E_0-semigroup can be obtained from the boundary weight map of a CP-flow acting on B(K \otimes L^2(0, \infty)) for some separable Hilbert space K. In this paper, we define boundary weight maps through boundary weight doubles (ϕ, ν), where ϕ: M_n(\C) \to M_n(\C) is a q-positive map and νis a boundary weight over L^2(0, \infty). These doubles induce CP-flows over K for 1 1, n \in \mathbb{N}.

math.OA↗

arXiv2010

Unital q-positive maps on M_2(\C) and a related E_0-semigroup result

From previous work, we know how to obtain type II_0 E_0-semigroups using boundary weight doubles (ϕ, ν), where ϕ: M_n(\C) \to M_n(\C) is a unital q-positive map and νis a normalized unbounded boundary weight over L^2(0, \infty). In this paper, we classify the unital q-positive maps ϕ: M_2(\C) \to M_2(\C). We find that every unital q-pure map ϕ: M_2(\C) \to M_2(\C) is either rank one or invertible. We also examine the case n=3, finding the limit maps L_ϕfor all unital q-positive maps ϕ: M_3(\C) \to M_3(\C). In conclusion, we present a cocycle conjugacy result for E_0-semigroups induced by boundary weight doubles (ϕ, ν) when νhas the form ν(\sqrt{I - Λ(1)} B \sqrt{I - Λ(1)})=(f,Bf).

math.FA↗