In this article, we address the following question: Is it true that the spatial numerical range (SNR) \(V_A(a)\) of an element a in a normed algebra \((A, \Vert \cdot \Vert )\) is always convex? If the normed algebra is unital, then it is convex (Bonsall and Duncan in Numerical ranges of operators on normed spaces and of elements of normed algebras, 1971, Theorem 3, P.16]). 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 \(V_A(a)\) is convex in several non-unital Banach algebras.
