On the geometry of the algebraic Davis--Wielandt shell and norm-parallelism in $C^{\ast }$-algebra

This article is devoted to the study of the Davis–Wielandt shell and the Davis–Wielandt radii of elements in a $C^{\ast }$-algebra. Utilizing a state-space approach, several geometric properties of the algebraic Davis–Wielandt shell are established. Upper and lower bounds for the algebraic Davis–Wielandt radii are obtained including the Davis–Wielandt radius of the sum of $k$ elements. We also explore the relationship between norm-parallelism and the Davis–Wielandt radii of elements.