In recent years, the theory of numerical range, developed in [3], has provided techniques which have considerably simplified the proofs of certain results in the theory of B*-algebras. The following question, posed by G. Lumer at the North British Functional Analysis Seminar held at Edinburgh in April 1968, is, therefore, natural: Can one prove the above theorem of Kadison using the techniques of the theory of numerical range? Lumer showed that such a proof can be given when the algebras concerned are commutative. In this paper, we give a simple, intrinsic proof of Kadison's result, using certain elementary notions from the theory of numerical range. We note that if A is a B*-algebra with identity 1, the set