The Spatial Numerical Range in Non-unital, Normed Algebras and Their Unitizations
Let (A,\|·\|) be any normed algebra (not necessarily complete nor unital). Let a A and let V_A(a) denote the spatial numerical range of a in (A,\|·\|). Let A_e = A + C1 be the unitization of A. If A is faithful, then we get two norms on A_e; namely, the operator norm \|·\|_op and the _1-norm \|·\|_1. Let A_op = (A,\|·\|_op ), A_op,e = (A_e,\|·\|_op ), and A_1,e = (A_e,\|·\|_1). We can calculate the spatial numerical range of a in all three normed algebras. Because the spatial numerical range highly depends 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. Some results that are proved in ref2, Section 2, and ref3, Section 10, will become corollaries of our results. We shall also show that the completeness and regularity of the norm is not required in ref6, Theorem 2.3. , number=2, journal=Sarajevo Journal of Mathematics, author=Dedania, Haresh V. and Patel, Avadh B., year=2025, month=Apr., pages=255–262