On the classification and modular extendability of E0‐semigroups on factors

Panchugopal Bikram and Daniel Markiewicz

Mathematische Nachrichten, 2020, vol. 293, issue 7, 1228-1250

Abstract: In this paper we study modular extendability and equimodularity of endomorphisms and E0‐semigroups on factors, when these definitions are recast to the context of faithful normal semifinite weights and this dependency is analyzed. We show that modular extendability is a property that does not depend on the choice of weights, it is a cocycle conjugacy invariant and it is preserved under tensoring. Furthermore, we prove a necessary and sufficient condition for equimodularity of endomorphisms in the context of weights. This extends previously known results regarding the necessity of this condition in the case of states. The classification of E0‐semigroups on factors is considered: a modularly extendable E0‐semigroup is said to be of type EI, EII or EIII if its modular extension is of type I, II or III, respectively. We prove that all types exist on properly infinite factors. We show that q‐CCR flows are not extendable, and we extend previous results by the first author regarding the non‐extendability of CAR flow to a larger class of quasi‐free states. We also compute the coupling index and the relative commutant index for the CAR flows and q‐CCR flows. As an application, by considering repeated tensors of the CAR flows we show that there are infinitely many non cocycle conjugate non‐extendable E0‐semigroups on the hyperfinite factors of types II1 and IIIλ, for λ∈(0,1).

Date: 2020
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.201800552

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:293:y:2020:i:7:p:1228-1250

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().