On Injective Envelopes of AF-Algebras

Ali Mahmoodi, Mohammad Reza Mardanbeigi

Abstract

In this paper, we prove that in the category of C*-algebras and completely positive linear maps an injective AF-algebra must be finite dimensional. We also show that a separable essentially simple C*-algebra whose injective envelopeis a von Neumann algebra must be an AF-algebra. Also, we show that if the regular completion (or equivalently, the injective envelope) of an essentially simple AF-algebra is  a W*-algebra, then the AF-algebra is isomorphic to a direct sum of elementary C*-algebras.