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

Panchugopal Bikram, Daniel Markiewicz

Research output: Contribution to journalArticlepeer-review


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 (Formula presented.).

Original languageEnglish
Pages (from-to)1228-1250
Number of pages23
JournalMathematische Nachrichten
Issue number7
StatePublished - 1 Jul 2020


  • CAR flows
  • CCR flows
  • E-semigroups
  • coupling index
  • modularly extendable endomorphisms
  • q-CCR flows
  • relative commutant index

ASJC Scopus subject areas

  • General Mathematics


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