STUDIA MATHEMATICA 153 (2) (2002) Weak Closure Theorem fails for Z 2 -actions by T. Downarowicz (Wrocław) and J. Kwiatkowski (Toruń) Abstract. We construct an example of a Morse Z 2 -action which has rank one and whose centralizer contains elements which cannot be weakly approximated by the trans- formations of the action. Introduction. The rank of a measure preserving transformation (i.e., of a Z-action) was defined by B. V. Chacon in 1970 ([C]). Roughly speaking, rank is the minimal number of towers needed to approximate the sigma- algebra, and it is an invariant of measure conjugacy. Chacon proved that rank is always not smaller than spectral multiplicity, which makes rank an important parameter. Whether rank is an invariant of spectral isomorphism is a famous, still unsolved, problem in ergodic theory. Rank can also be defined for actions of other abelian groups whenever it is clear what we understand by towers. This is equivalent to a choice of a F¨ olner sequence of sets. For Z 2 the most natural choice is rectangles (or parallelepipeds in Z d ). With such choice Chacon’s inequality still holds. The metric centralizer of an abelian group T of measure preserving trans- formations is defined as the family of all automorphisms of the space which commute with all elements of T. Endowed with the weak topology, it has the structure of a complete metric topological group and obviously it contains T, hence also its closure Wcl(T). The centralizer can be viewed as a family of special self-joinings and hence has gained a lot of attention in ergodic theory (see e.g. [K2], [J-R]). In 1986 J. King ([K1]) proved that for rank-one Z-actions the centralizer coincides with Wcl(T) (see also [R] for another proof). This theorem is 2000 Mathematics Subject Classification : 37A05, 37A15, 37A35. Key words and phrases : Z 2 -action, weak closure theorem, rank one, metric centralizer, Morse sequence. Research of the second author supported by the grant KBN 5 P03A 027 21. The first draft of this paper was written while the authors were visiting the Universit´ e de Bretagne Occidentale, March 2001. [115]