The commutant is the weak closure of the powers, for rank-1 transformations

Abstract

Abstract. In the class of rank-1 transformations, there is a strong dichotomy. For such a T, the commutant is either trivial, consisting only of the powers of T, or is uncountable. In addition, the commutant semigroup, C{T), is in fact a group. As a consequence, the notion of weak isomorphism between two transformations is equivalent to isomorphism, if at least one of the transformations is rank-1. In § 2, we show that any proper factor of a rank-1 must be rigid. Hence, neither Ornstein's rank-1 mixing nor Chacon's transformation, can be a factor of a rank-1. N.B. In this paper 'transformation' shall mean a measure preserving transformation on a Lebesgue probability space. In general, we assume our measure spaces nonatomic; however the spaces that factors live on may be atomic. A transformation T is invertible if T" 1 exists and is a transformation. All sets mentioned are assumed measurable. In particular, S will be invertible if the T t are various powers of a single transformation.

Extensions of generic measure-preserving actions

“…Conversely, the next proposition shows that, when Y is separable, the existence of points of local density is necessary for f (Y ) to be non meager, though it is certainly not necessary that the set of points of local density be dense; as a side remark, note that this is always a G δ subset of Y , see [13]. …”

“…In an earlier version of this article, the above result was incorrectly attributed to King [13]; actually, it was proved much earlier: it is stated in [4], where the authors say it was already proved by Chacon-Schwartzbauer [5] (though the result does not seem to appear explicitly there) (3) . Yet another proof recently appeared in [15].…”

“…[13,14,19,20]); below we quickly discuss this approach, as well as the technique used in [15], and compare the two. Proof.…”

Weakly mixing rank-one transformations conjugate to their squares

“…As an application, we show that for any n, there exists a weakly mixing transformation T conjugate to T 2 and such that the rank of T is finite and greater than n (Theorem 4.10). In [Go2] Goodson isolated the class of those T (conjugate to T 2 ) that have King's weak closure property [Ki1]. Such T do not commute with periodic transformations of even order.…”

“…Indeed, if V (1,0) were of rank one then its factor T × T 2 would also be of rank one. Hence by King's weak closure theorem [Ki1] the transformation Id ×T commuting with T × T 2 is the weak limit of a sequence of powers of T × T 2 . Thus Id = lim i→∞ T n i and T = lim i→∞ T 2n i , a contradiction.…”