The local multiplier algebra of a C∗-algebra with finite dimensional irreducible representations

We describe, up to isomorphism, the local multiplier algebra Mloc(A) of a C∗-algebra A which admits only finite dimensional irreducible representations: Mloc(A) in this case is a finite or countable direct product of C∗-algebras of the form C(Xn)⊗Mn, where each space Xn is Stonean. In particular, Mloc(A) is an AW∗-algebra of type I, so it admits only inner derivations and it coincides with the injective envelope I(A) of A.