In this article, we address the following question: Is it true that the
spatial numerical range (SNR) VAβ(a) of an element a in a normed algebra
(A,β₯β β₯) is always convex? If the normed algebra is unital, then it is
convex \cite[Theorem 3, P.16]{BoDu:71}. In non-unital case, we believe that the
problem is still open and its answer seems to be negative. In search of such a
normed algebra, we have proved that the SNR VAβ(a) is convex in several
non-unital Banach algebras.Comment: 9 page