One-parameter groups of orthogonality preservers on C$$^*$$-algebras

Abstract

We establish a more precise description of those surjective or bijective continuous linear operators preserving orthogonality between C\(^*\)-algebras. The new description is applied to determine all uniformly continuous one-parameter semigroups of orthogonality preserving operators on an arbitrary C\(^*\)-algebra. We prove that given a family \(\{T_t: t\in {\mathbb {R}}_0^{+}\}\) of orthogonality preserving bounded linear bijections on a general C\(^*\)-algebra A with \(T_0=Id\), if for each \(t\ge 0,\) we set \(h_t = T_t^{**} (1)\) and we write \(r_t\) for the range partial isometry of \(h_t\) in \(A^{**},\) and \(S_t\) stands for the triple isomorphism on A associated with \(T_t\) satisfying \(h_t^* S_t(x)\) \(= S_t(x^*)^* h_t\), \(h_t S_t(x^*)^* =\) \( S_t(x) h_t^*\), \(h_t r_t^* S_t(x) =\) \(S_t(x) r_t^* h_t\), and \(T_t(x) = h_t r_t^* S_t(x) = S_t(x) r_t^* h_t, \hbox { for all } x\in A,\) 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 \( h_{t+s} = h_t r_t^* S_t^{**} (h_s),\) holds for all \(s,t\in {\mathbb {R}}.\)