Injective Hulls of C* Algebras. Ii
نویسنده
چکیده
Proof. The idempotents correspond to the Borel sets modulo sets of first category. Since, in addition, the idempotents generate B(X), B(X) is an AW* and hence an injective algebra. The natural map U of C(X) into B(X) induced by the inclusion map is clearly a homomorphism. It is one-one since continuous functions which are not identically equal must differ on a set of second category. To complete the proof, it suffices by Theorem 6 in [2] to show that for every nonzero idempotent e in B(X) there is an element 0^/G C(X) such that e(Uf)= Uf. Since e is represented by a characteristic function on a regular open set, this follows immediately from Urysohn's lemma. Q.E.D. Remark. By avoiding the maximal ideal spaces this theorem brings the subject closer to classical analysis. The relationship with the
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