One-parameter groups of orthogonality preservers on C${}^{\ast }$-algebras

Abstract

We establish a more precise description of those surjective or bijective continuous linear operators preserving orthogonality between C${}^{\ast }$-algebras. The new description is applied to determine all uniformly continuous one-parameter semigroups of orthogonality preserving operators on an arbitrary C${}^{\ast }$-algebra. We prove that given a family $\{T_t:t\in {\mathbb{R}}_0^{+}\}$ of orthogonality preserving bounded linear bijections on a general C${}^{\ast }$-algebra $A$ with $T_0=Id$, if for each $t\ge 0,$ we set $h_t=T_t^{\ast \ast }(1)$ and we write $r_t$ for the range partial isometry of $h_t$ in $A^{\ast \ast },$ and $S_t$ stands for the triple isomorphism on $A$ associated with $T_t$ satisfying $h_t^{\ast }{S}_t(x)$ $=S_t(x^{\ast }{)}^{\ast }{h}_t$, $h_t{S}_t(x^{\ast }{)}^{\ast }=$ $S_t(x)h_t^{\ast }$, $h_t{r}_t^{\ast }{S}_t(x)=$ $S_t(x)r_t^{\ast }{h}_t$, and $T_t(x)=h_t{r}_t^{\ast }{S}_t(x)=S_t(x)r_t^{\ast }{h}_t,for\; allx\in A,$ the following statements are equivalent: $(a)$ $\{T_t:t\in {\mathbb{R}}_0^{+}\}$ is a uniformly continuous one-parameter semigroup of orthogonality preserving operators on $A$; $(b)$ $\{S_t:t\in {\mathbb{R}}_0^{+}\}$ is a uniformly continuous one-parameter semigroup of surjective linear isometries (i.e. triple isomorphisms) on $A$ (and hence there exists a triple derivation $\delta$ on $A$ such that $S_t=e^{t\delta }$ for all $t\in \mathbb{R}$), the mapping $t\mapsto h_t$ is continuous at zero, and the identity $h_{t+s}=h_t{r}_t^{\ast }{S}_t^{\ast \ast }(h_s),$ holds for all $s,t\in \mathbb{R}.$