On the comparison of norms of convolutors associated with noncommutative dynamics

Abstract

To any action of a locally compact group G on a pair (A, B) of von Neumann algebras is canonically associated a pair (π α A , π α B ) of unitary representations of G. The purpose of this paper is to provide results allowing to compare the norms of the operators π α A (µ) and π α B (µ) for bounded measures µ on G. We have a twofold aim. First to point out that several known facts in ergodic and representation theory are indeed particular cases of general results about (π α A , π α B ). Second, under amenability assumptions, to obtain transference of inequalities that will be useful in noncommutative ergodic theory.1991 Mathematics Subject Classification. Primary 46L55; Secondary 46L10, 22D25.

Spectral rigidity of group actions on homogeneous spaces

“…We prefer to call them co-amenable in order to avoid confusion with the well-established notion of an amenable action G X due to Zimmer ([Zimm84]), which in the case X = G/H corresponds to the amenability of H; a unification of both notions by means of an appropriate definition of amenable actions on pairs of measure spaces is given in [Zimm78]. For a further extension of these notions to the context on non-commutative measure spaces (that is, to the context of von Neumann algebras), see [Anan08]. (iii) If X is a locally compact space, one may define, as in [Gree69] or [Guiv80], co-amenability of the action G X through the existence of a G-invariant mean on the space C b (X) of continuous bounded functions on X; as Example 5.10 below shows, this is in general a weaker condition than co-amenability of the action of G on the measure space (X, m), even in the case of a homogeneous space X = G/H (for a group G containing G), where a natural quasi-invariant measure (class) m is given.…”

“…(i) Considering the action G G given by left translation, we see that Theorem 3.20 is a special case of Theorem 5.7. (ii) For an extension of Theorem 5.7 to amenable pair of actions (including actions on von Neumann algebras), see [Anan03] and [Anan08]).…”