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In 1988, Haagerup and Størmer conjectured that every pointwise inner automorphism of a type ${\rm III_1}$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that every type ${\rm III_1}$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and Størmer's conjecture holds in full generality if Connes' bicentralizer problem has an affirmative answer. Our proof is based on Popa's intertwining theory and Marrakchi's recent work on relative bicentralizers. [ABSTRACT FROM AUTHOR]

Bożejko and Speicher associated a finite von Neumann algebra MT to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter relation and $ \left\| T \right\|. We show that if dim $(\mathcal {H})$ ⩾ 2, then MT is a factor when T admits an eigenvector of some special form. [ABSTRACT FROM AUTHOR]

We show that every inclusion of von Neumann algebras with a faithful normal conditional expectation has the weak relative Dixmier property. This answers a question of Popa (J. Funct. Anal. 171 (2000) 139–154). The proof uses an improvement of Ellis's lemma for compact convex semigroups. [ABSTRACT FROM AUTHOR]

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 III 1 factor is always ergodic. As a consequence, for any type III 1 factor M and any λ ∈ ] 0 , 1 ] , there exists an irreducible AFD type III λ subfactor with expectation in M. Moreover, any type III 1 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 III 1 factor is approximately inner. [ABSTRACT FROM AUTHOR]

Let M be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes _\sigma \Gamma $. When Γ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes _\sigma \Gamma $ is full if and only if M is full and the quotient map $\overline {\sigma } : \Gamma \rightarrow {\rm out}(M)$ has finite kernel and discrete image. This answers the question of Jones from [11]. When M is full and Γ is arbitrary, we give a sufficient condition for $M \rtimes _\sigma \Gamma $ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if M is any full factor (possibly of type III) and Γ is a non-inner amenable group, then the crossed product $M \rtimes _\sigma \Gamma $ is full. [ABSTRACT FROM AUTHOR]