Deformation of involution and multiplication in a $C^*$-algebra
Volume 215 / 2013
Studia Mathematica 215 (2013), 31-37
MSC: Primary 46L05; Secondary 46L10.
DOI: 10.4064/sm215-1-3
Abstract
We investigate the deformations 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 when 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 the center of a unital $C^*$-algebra $\mathcal {A}$ to be trivial.
