Semiprojectivity with and without a group action

The equivariant version of semiprojectivity was recently introduced by the first named 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⁎C⁎-algebra A, then A   must be semiprojective. This fails for noncompact groups: we construct a semiprojective action of ZZ on a nonsemiprojective C⁎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⁎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⁎C⁎-algebra is a semiprojective C⁎C⁎-algebra. We give examples to show that this does not generalize to all compact groups.