The equivariant version of semiprojectivity was recently introduced by the
first author. We study properties of this notion, in particular its relation to
ordinary semiprojectivity of the crossed product and of the algebra itself.
We show that equivariant semiprojectivity is preserved when the action is
restricted to a cocompact subgroup. Thus, if a second countable compact group
acts semiprojectively on a C*-algebra A, then A must be semiprojective.
This fails for noncompact groups: we construct a semiprojective action of the
integers on a nonsemiprojective C*-algebra.
We also study equivariant projectivity and obtain analogous results, however
with fewer restrictions on the subgroup. For example, if a discrete group acts
projectively on a C*-algebra A, then A must be projective. This is in
contrast to the semiprojective case.
We show that the crossed product by a semiprojective action of a finite group
on a unital C*-algebra is a semiprojective C*-algebra. We give examples to show
that this does not generalize to all compact groups.Comment: 38 page
