Geodesic
Some C*-Algebras with Outer Derivations, II
Canadian journal of mathematics, Tome 26 (1974) no. 1, pp. 185-189

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In this paper we shall consider the class of C*-algebras which are inductive limits of sequences of finite-dimensional C*-algebras. We shall give a complete description of those C*-algebras in this class every derivation of which is inner.Theorem. Let A be a C*-algebra. Suppose that A is the inductive limit of a sequence of finite-dimensional C*-algebras. Then the following statements are equivalent: (i) every derivation of A is inner; (ii) A is the direct sum of a finite number of algebras each of which is either commutative, the tensor product of a finite-dimensional and a commutative with unit, or simple with unit. 
Elliott, George A. Some C*-Algebras with Outer Derivations, II. Canadian journal of mathematics, Tome 26 (1974) no. 1, pp. 185-189. doi: 10.4153/CJM-1974-018-6
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