Property $L$ and asymptotic abelianness for $W^*$-algebras

Abstract:

(i) Let $\mathcal {A}$ be a finite $W^*$-algebra acting on a separable Hilbert space and having no abelian direct summand. If $\mathcal {A}$ is asymptotically abelian, then $\mathcal {A}$ has property L. (ii) Let $\mathcal {A}$ be a finite $W^*$-algebra acting on a separable Hilbert space. Then $\mathcal {A} \otimes B(h), h$ a separable infinite dimensional Hilbert space, is not asymptotically abelian. (iii) Type $\mathrm {II}_\infty W^*$-algebras are not asymptotically abelian. (iv) Noncommutative type I $W^*$-algebras are not asymptotically abelian. (v) The type III factor $\mathcal {B} = \mathcal {P} \otimes \mathcal {A}(G)$ is not asymptotically abelian. $\mathcal {B}$ produces uncountably many nonisomorphic nonasymptotically abelian factors of type III and establishes an example of a purely infinite factor that has property L but is not asymptotically abelian.