On the geometry of the algebraic Davis--Wielandt shell and norm-parallelism in $C^*$-algebra

This article is devoted to the study of the Davis--Wielandt shell and the Davis--Wielandt radii of elements in a $C^*$-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.