On Injective Envelopes of G-AF-algebras

Let G be a discrete group that acts on C^*-algebra A. We prove that in the category of G-AF-algebras and completely positive linear G-equivariant the finite-dimensional G-AF-algebra must be injective. One consequence of this is that no infinite-dimensional G-AF-algebra could be injective in the category of G-C^∗-algebras.Also, we show that the following statements are equivalent for a separable essentially simple G-C^*-algebra A: (i) A̅_G, regular completion of A, is a G-W^*-algebra (von-Neumann algebra).(ii) I_G(A), injective envelope of A, is a G-W*-algebra (von-Neumann algebra).(iii) A is G-isomorphic to a direct sum of elementary G-C^*-algebras K(H_n), which H_n is a Hilbert space. Since the K(H_n) is G-AF-algebra, and we know that the category of G-AF-algebras is closed under taking the countable direct sum. Therefore, we prove this G-C^*-algebra is a G-AF-algebra.Further, we show that if separable essentially simple G-C^∗-algebra A is post liminal, then A is liminal G-C^∗-algebra.