Spatial Numerical Range in Non-unital, Normed algebras and their Unitizations

Let $(A, \|\cdot\|)$ be any normed algebra (not necessarily complete nor unital). Let $a \in A$ and let $V_A(a)$ denote the spatial numerical range of $a$ in $(A, \|\cdot\|)$. Let $A_e = A + {\mathbb C} 1$ be the unitization of $A$. If $A$ is faithful, then we get two norms on $A_e$; namely, the operator norm $\|\cdot\|_{op}$ and the $\ell^1$-norm $\|\cdot\|_1$. Let $A^{op} = (A, \|\cdot\|_{op})$, $A_e^{op} = (A_e, \|\cdot\|_{op})$, and $A_e^1 = (A_e, \|\cdot\|_1)$. We can calculate the spatial numerical range of $a$ in all these three normed algebras. Because the spatial numerical range highly depend on the identity as well as on the completeness and the regularity of the norm, they are different. In this paper, we study the relations among them. Most of the results proved in \cite{BoDu:71, BoDu:73} will become corollaries of our results. We shall also show that the completeness and regularity of the norm is not required in \cite[Theorem 2.3]{GaHu:89}.