Answering a question of A. Vershik I will present a
construction of two non-weakly isomorphic ergodic automorphisms
for which the associated unitary (Koopman) representations are
Markov quasi-similar. I will also discuss metric invariants of
Markov quasi-similarity in the class of ergodic automorphisms. My
talk is based on a joint paper with M. LemaĆczyk.