论文标题
c $^*$ - 代数的正交性保存器的单参数组
One-parameter groups of orthogonality preservers on C$^*$-algebras
论文作者
论文摘要
我们建立了那些过滤式或两种冲突的连续线性操作员的更精确的描述，该算子保存了C $^*$ - 代数之间的正交性。新的描述用于确定在任意C $^*$ - 代数上保存正交操作员的所有均匀连续的单参数半群。我们证明，给定的一个家庭$ \ {t_t：t \ in \ mathbb {r} _0^{+} \} $的正交性保留有限的线性线性在一般C $^*$ - 代数 - 代数 - 代数$ -T_0 = ID $的$ t_0 = ID $，如果$ t_0我们为$ h_t $ in $ a^{**}，$和$ s_t $ in $ h_t $的范围部分等轴测编写$ r_t $，与满足$ h_t^*s_t^*s_t^*s_t（x）$ $ t_t $相关的$ a $ a $ a $ a $ a $ s. h_t^* $，$ h_t r_t^* s_t（x）= $ $ s_t（x）r_t^* h_t $，和$ t_t（x）= h_t r_t r_T^* s_t（x）= s_t（x）= s_t（x）r_t^* h_t^* h_t^* h_t h_t，for All} $（a）$ $ \ {t_t：t \ in \ mathbb {r} _0^{+} \} $是一个均匀连续的连续连续的单参数semigroup，在$ a $ a $ a $ a $ a;; $（b）$ $ \ {s_t：t \ in \ mathbb {r} _0^{+} \} $是一个均匀连续的连续连续的单参数线性异构体（即三重异形）（即三倍异态）的$ a $ a $ a $ a $ a $ n $ a $ a $ a $ a $ a $ a δ}$ 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 $s,t\in \mathbb{R}.$
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 $t\geq 0,$ 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 $δ$ on $A$ such that $S_t = e^{t δ}$ 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 $s,t\in \mathbb{R}.$
