Deformation of involution and multiplication in a C*-algebra

Abstract

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

8 pages, to appear in Studia Math. (IMPAN)

References in corpus (1)

• A characterisation of C*-algebras through positivity of functionals