In the past two decades, Sorin Popa’s breakthrough deformation/rigidity
theory has produced remarkable rigidity results for von Neumann algebras
which can be deformed inside a larger algebra
by an action
, while simultaneously
containing subalgebras
rigid with respect to that deformation, that is, such that
uniformly on
the unit ball of
as
.
However, it has remained unclear how to exploit the interplay between distinct rigid
subalgebras not in specified relative position.
We show that in fact, any diffuse subalgebra which is rigid with respect
to a mixing s-malleable deformation is contained in a subalgebra which is
uniquely maximal with respect to being rigid. In particular, the algebra
generated by any family of rigid subalgebras that intersect diffusely must itself
be rigid with respect to that deformation. The case where this family has
two members was the motivation for this work, showing for example that if
is a countable
group with
,
then
cannot be generated by two property-(T) subalgebras with diffuse intersection;
however, the result is most striking when the family is infinite.
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