论文标题
与交换性JB $^*$ - 三元组相关的功能空间之间的线性正交性保存器
Linear orthogonality preservers between function spaces associated with commutative JB$^*$-triples
论文作者
论文摘要
据Gelfand理论所知，每个交换性的JB $^*$ - 三重 - 三重表示表示形式的连续函数的空间，$ c_0^{\ Mathbb {\ Mathbb {t}}}}（l）= \ {a \ {a \ {a \ in C_0（l）中： t \ in l \}，$$其中$ l $是主要$ \ mathbb {t} $ - 捆绑包和$ \ mathbb {t} $表示$ \ mathbb {c}中的单位圆。我们表明，每个线性正交性保留$ t：c_ {0}^{\ Mathbb {t}}}（l_1）\ to C_ {0}^{\ MathBB {t Mathbb {t}}（t}}（t}}（l_2）$在其图像上以$ y的图像为$ potimy opity potime potibort $ n offort $ $ t $ VANISHES，以及这些点$ S $在$ L_2 $中形成的第三部分，因此评估映射$δ_s\ circe \ circt t $是不连续的。在此表示的后果之一中，我们得到的是，在交换性JB $^*$之间保存正交性的每个线性双线都可以自动连续并保留生物表达性。
It is known, by Gelfand theory, that every commutative JB$^*$-triple admits a representation as a space of continuous functions of the form $$C_0^{\mathbb{T}}(L) = \{ a\in C_0(L) : a(λt ) = λa(t), \ \forall λ\in \mathbb{T}, t\in L\},$$ where $L$ is a principal $\mathbb{T}$-bundle and $\mathbb{T}$ denotes the unit circle in $\mathbb{C}.$ We provide a description of all orthogonality preserving (non-necessarily continuous) linear maps between commutative JB$^*$-triples. We show that each linear orthogonality preserver $T: C_{0}^{\mathbb{T}} (L_1)\to C_{0}^{\mathbb{T}} (L_2)$ decomposes in three main parts on its image, on the first part as a positive-weighted composition operator, on the second part the points in $L_2$ where the image of $T$ vanishes, and a third part formed by those points $s$ in $L_2$ such that the evaluation mapping $δ_s\circ T$ is non-continuous. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB$^*$-triples is automatically continuous and biorthogonality preserving.
