We show that a factor M is full if and only if the $$C^*$$ C ∗ -algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type $$\mathrm{III}_1$$ III 1 factor is always ergodic. As a consequence, for any type $$\mathrm{III}_1$$ III 1 factor M and any $$\lambda \in ]0,1]$$ λ ∈ ] 0 , 1 ] , there exists an irreducible AFD type $$\mathrm{III}_\lambda $$ III λ subfactor with expectation in M . Moreover, any type $$\mathrm{III}_1$$ III 1 factor M which satisfies $$M \cong M \mathbin {{\overline{\otimes }}}R_\lambda $$ M ≅ M ⊗ ¯ R λ for some $$\lambda \in ]0,1[$$ 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 $$\mathrm{III}_1$$ III 1 factor is approximately inner.

中文翻译：

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全因子、双中心化流和近似内自同构

我们证明了一个因子 M 是满的，当且仅当由其左右正则表示生成的 $$C^*$$ C ∗ -代数包含紧致算子。我们证明了 $$\mathrm{III}_1$$ III 1 因子类型的双中心化流总是遍历的。因此，对于任何类型 $$\mathrm{III}_1$$ III 1 factor M 和任何 $$\lambda \in ]0,1]$$ λ ∈ ] 0 , 1 ] ，存在不可约 AFD 类型$$\mathrm{III}_\lambda $$ III λ 子因子，在 M 中具有期望。此外，任何类型 $$\mathrm{III}_1$$ III 1 factor M 满足 $$M \cong M \mathbin {{\overline{\otimes }}}R_\lambda $$ M ≅ M ⊗ ¯ R λ对于某些 $$\lambda \in ]0,1[$$ λ ∈ ] 0 ， 1 [ 具有微不足道的双中心化器。最后，我们给出了 Connes 猜想的近似内自同构的表征的反例，我们证明了这个猜想的较弱版本。