Deformation of involution and multiplication in a C*-algebra

TL;DR

This paper characterizes all possible deformations of involution and multiplication in a unital C*-algebra that preserve its norm, and explores conditions for trivial centers and positivity of elements.

Contribution

It provides a complete description of all involutions and multiplications maintaining the C*-algebra structure with fixed norm, and introduces methods to make any invertible element positive.

Findings

Classifies all deformations of involution and multiplication preserving C*-structure.

Provides necessary and sufficient conditions for trivial centers in unital C*-algebras.

Introduces a way to turn any invertible element into a positive element within a deformed C*-algebra.

Abstract

We investigate the deformation of involution and multiplication in a unital $C^*$-algebra when its norm is fixed. Our main result is to present all multiplications and involutions on a given $C^*$-algebra $\mathcal{A}$ under which $\mathcal{A}$ is still a $C^*$-algebra whereas we keep the norm unchanged. For each invertible element $a\in\mathcal{A}$ we also introduce an involution and a multiplication making $\mathcal{A}$ into a $C^*$-algebra in which $a$ becomes a positive element. Further, we give a necessary and sufficient condition for that the center of a unital $C^*$-algebra $\mathcal{A}$ is trivial.