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

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