Let ~ ' be a semi-simple commutative convolution measure algebra as described by Taylor [9]. The maximal ideal space A of ~ ' is representable as the semigroup of continuous semicharacters on a compact semigroup the structure semigroup of J/g. Using this description of A, Taylor has shown that the strong boundary points he A of ~ ' must have idempotent modulus (Ihl 2 = 1hi). On the other hand it is relatively easy to exhibit idempotents which are not strong boundary points in fact, which are not even in the Silov boundary. For example, let 2 /cons i s t of all summable sequences (a,),__> o with convolution as multiplication. Then the idempotent complex homomorphism (a~),,~a o is not in the Silov boundary of ~ ' . For the most important example, where ~ ' =M(G), the convolution algebra of finite regular complex valued Borel measures on a locally compact abelian group G, the question of whether idempotents are strong boundary points is less trivial. In fact the question of whether all idempotents are in the Silov boundary of M(G) remains unresolved. However, it is known (see [3]) that if the idempotent h in question corresponds to the direct sum decomposition of M(G) produced by a single generator symmetric Raikov system then there is an analytic disc around h and this can be used to show that h is not a strong boundary point. To be precise the disc is analytic not only for M(G) but for the uniform closure M(G~'-of the algebra of Gelfand transforms on A, and this prevents h being a strong boundary point. As a simple example, if the Raikov system consists of all countable subsets of G, then h is the complex homomorphism/~,~ S d#n where #n is the discrete part