Full factors, bicentralizer flow and approximately inner automorphisms

2018/11/26 by Marrakchi, Amine
#46L10 #46L36 #46L40 #46L55 #FOS: Mathematics #Operator Algebras (math.OA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1811.10253

Abstract

We show that a factor M is full if and only if the C^*-algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type III₁ factor is always ergodic. As a consequence, for any type III₁ factor M and any λ∈ ]0,1], there exists an irreducible AFD type III_λ subfactor with expectation in M. Moreover, any type III₁ factor M which satisfies M ≅ M ⊗ R_λ for some λ∈ ]0,1[ has trivial bicentralizer. Finally, we give a counter-example to the characterization of approximately inner automorphisms conjectured by Connes and we prove a weaker version of this conjecture. In particular, we obtain a new proof of Kawahigashi-Sutherland-Takesaki's result that every automorphism of the AFD type III₁ factor is approximately inner.

Related

• Structure of bicentralizer algebras and inclusions of type III factors
• Haagerup and Størmer's conjecture for pointwise inner automorphisms
• Kadison's problem for type III subfactors and the bicentralizer conjecture
• Unique prime factorization and bicentralizer problem for a class of type III factors
• Pointwise inner automorphisms of almost periodic factors