On the Closure of the Numerical Range of an Operator

If T is a bounded linear mapping (briefly, operator) in a Hilbert space 3C, the numerical range of T is the set WiT) = {(Tx, x): |[x|| = l}; thus WiT) is convex [8, p. 131], and its closure clfW^r)] is compact and convex. Roughly speaking, in this note we observe that cl[TF(T)] can be uniquely defined for an element T of an abstract C*-algebra, while WiT) cannot. The C*-algebra setting yields an extension of the spectral convexity theorem [8, p. 327] to nonnormal operators (Corollary 1 of Theorem 3), as well as a reformulation of a theorem of C. Berger (Corollary 2 of Theorem 3).