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Characters on C*-algebras, the joint normal spectrum, and a pseudo-differential C*-algebra

Let {Ai; i ∈ Ω} be a family of C*-algebras acting on a Hubert space H and let A be the C*-algebra that they generate. We shall assume throughout that C*-algebras always contain the identity operator. Let M(A) denote the space of characters, that is, multiplicative linear functionals, acting on A, with the weak star topology. We obtain here a natural characterisation of M(A) as a subset of the product space determined by {M(Ai); i ∈ Ω}. In the case of singly generated C*-algebras this characterisation is related to the joint normal spectrum (5) of a family of operators.