Uniqueness of equivariant pseudo-expectations from crossed-product simplicity

Determine whether simplicity of the reduced crossed product A\rtimes_r\Gamma of a unital \Gamma-simple \Gamma-C^*-algebra A forces the canonical map A\rtimes_r\Gamma\to I_\Gamma(A) to be the unique \Gamma-equivariant pseudo-expectation.

Background

The paper studies generalized Powers averaging for reduced crossed products A\rtimes_r\Gamma, where A is a unital \Gamma-simple C^*-algebra and I_\Gamma(A) is its \Gamma-equivariant injective envelope. Its main theorem proves that generalized Powers averaging is equivalent to uniqueness of the canonical \Gamma-equivariant pseudo-expectation from A\rtimes_r\Gamma to I_\Gamma(A).

The authors distinguish this uniqueness property from simplicity of the crossed product. Although simplicity is equivalent to faithfulness of every equivariant pseudo-expectation, the paper does not establish that simplicity alone implies uniqueness. It instead gives sufficient additional conditions—vanishing obstruction and FC-hypercentrality—for simplicity to imply generalized Powers averaging and hence uniqueness.