JOINT NUMERICAL RANGES IN NON UNITAL NORMED ALGEBRAS

Let A denote a unital normed algebra over a field K = R or C and let e be the identity of A. Given $a \in A$ and $x \in A$ with $\Vert x \Vert = 1$, let $$ V(A, a, x) = {f(ax) : f \in A', f(x) = 1 = \Vert f \Vert}. $$ Then the (Bonsall and Duncan) numerical range of an element $a \in A$ is defined by $$ V(a) = \cup{V(A, a, x) : x \in A, \Vert x \Vert = 1}, $$ where A' denotes the dual of A. In [2], $V(a) = {f(a) : f \in A', f(e) = 1 = \Vert f \Vert}$.