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We prove a correct assertion on Property L for von Neumann algebras of type I: a type I von Neumann algebra M on a separable Hilbert space has Property L if and only if M contains no minimal projection. Furthermore, a correct proof of an assertion on asymptotic abelianness for von Neumann algebras of type I is also given. Let M be a von Neumann algebra on a separable Hilbert space and let U(M) represent the set of unitary operators in M. A uniformly bounded sequence {AJ} in M is said to be central if {AA, A"A } converges strongly to zero for each A in M. Recall that M is said to have Property L if there is a central sequence { U"} in U(M) such that weak limit U}, = 0. Moreover M is said to be asymptotically abelian if there is a sequence of *-automorphisms {?a} on M such that {a"(A)} is central for each A in M. In [2, Theorem 3], Sarian stated that type I von Neumann algebras do not have Property L. However, this theorem does not hold in general. Indeed, we can easily give a counter example. Let T be the unit circle and ML(T) be the algebra of all multiplications 7T(J) on L2(T) by essentially bounded Lebesgue measurable functions f on T. We put U,(t) = e" (0 < t < 27T). Then the sequence {fT(U")} in U(ML.(T) converges weakly to zero by the Riemann-Lebesgue Theorem. Since ML.(n is abelian, {fT(U,,)) is, of course, central. Thus ML(T) is a type I von Neumann algebra which has Property L. In the present paper, we give a necessary and sufficient condition for von Neumann algebras of type I to have Property L. In [3, Corollary 6], Sarian showed that if M is a finite type I von Neumann algebra having no abelian direct summand, then M is not asymptotically abelian. His proof, however, depends on the implication that if M is asymptotically abelian then it has Property L, which cannot be applied to the present case because of the above situation. Hence we have, independent of Property L, that nonabelian von Neumann algebras of type I are not asymptotically abelian. 1. Property L for von Neumann algebras of type I. We note first the relation between Property L and typical von Neumann algebras of type I: ML(T) and B(H) (the full operator algebra on a separable Hilbert space H). As we have stated in the introduction, we have the following. Received by the editors April 3, 1981 and, in revised form, May 6, 1981. 1980 Mathematics Subject Classification Primary 46L10.

Property 𝐿 and asymptotic abelianness for von Neumann algebras of type 𝐼