Abstract
In this paper, the notion of spatial numerical range of elements of Banach
algebras without identity is studied. Specifically, the relationship between spatial
numerical ranges, numerical ranges and spectra is investigated. Among other results,
it is shown that the closure of the spatial numerical range of an element of a Banach
algebra without Identity but wlth regular norm is exactly its numerical range as an
element of the unitized algebra. Futhermore, the closure of the spatial numerical
range of a hermitian element coincides with the convex hull of its spectrum. In
particular, spatial numerical ranges of the elements of the Banach algebra