Property 𝐿 and asymptotic abelianness for 𝑊*-algebras

Abstract

(i) LetA\mathcal {A}be a finiteW∗W^*-algebra acting on a separable Hilbert space and having no abelian direct summand. IfA\mathcal {A}is asymptotically abelian, thenA\mathcal {A}has propertyL. (ii) LetA\mathcal {A}be a finiteW∗W^*-algebra acting on a separable Hilbert space. ThenA⊗B(h),h\mathcal {A} \otimes B(h), ha separable infinite dimensional Hilbert space, is not asymptotically abelian. (iii) TypeII∞W∗\mathrm {II}_\infty W^*-algebras are not asymptotically abelian. (iv) Noncommutative type IW∗W^*-algebras are not asymptotically abelian. (v) The type III factorB=P⊗A(G)\mathcal {B} = \mathcal {P} \otimes \mathcal {A}(G)is not asymptotically abelian.B\mathcal {B}produces uncountably many nonisomorphic nonasymptotically abelian factors of type III and establishes an example of a purely infinite factor that has propertyLbut is not asymptotically abelian.