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On the classification and modular extendability of E 0 ‐semigroups on factors
Author(s) -
Bikram Panchugopal,
Markiewicz Daniel
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800552
Subject(s) - mathematics , endomorphism , semigroup , pure mathematics , centralizer and normalizer , context (archaeology) , type (biology) , conjugacy class , modular design , invariant (physics) , discrete mathematics , computer science , paleontology , ecology , mathematical physics , biology , operating system
In this paper we study modular extendability and equimodularity of endomorphisms and E 0 ‐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 E 0 ‐semigroups on factors is considered: a modularly extendable E 0 ‐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 E 0 ‐semigroups on the hyperfinite factors of types II 1 and III λ , for λ ∈ ( 0 , 1 ) .