论文标题
在Unital Jordan-Banach代数
Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras
论文作者
论文摘要
令$ m $和$ n $为Unital Jordan-Banach代数，让$ m^{ - 1} $和$ n^{ - 1} $分别表示$ m $和$ n $的可逆元素集。假设$ \ mathfrak {m} \ subseteq m^{ - 1} $和$ \ mathfrak {n} \ subseteq n^{ - 1} $是$ m^{ - 1} $和$ n^{ - 1} $的clopen子集，分别为$ n of powers和a）$ {$ {$ n of。在本文中，我们证明，对于每个过滤式等轴测$δ：\ Mathfrak {m} \ to \ mathfrak {n} $，存在一个过滤的实时等距$ t_0：m \ t_0：m \ to n $和元素$ u_0 $ u_0 $在$ n $ $ n $ $Δ（a）中$Δ（a）= a）= a）= t t t_0（a） \ Mathfrak {M} $。\ Smallskip 假设$ m $和$ n $是Unital jb $^*$ - 代数，我们确定每个过滤等轴测$δ：\ mathfrak {m} {m} \ to \ mathfrak {n} $ element $δ（\ \ textbf {1}） $^*$ - $ j $ $ j $从$ m $ to $ u^*$ - homotope $ n_ {u^*} $，以便$$δ（a）= j（p \ circe a） + j（（（（\ textbf {1} -p）-p）-p）\ circe a^*），$ a \ in $ a \ in \ in \ in \ nath \ nath \ nath \ nath \ nath \ mathfrak} $} $}。在其他假设的情况下，满足$ u_ {ω_0}}（δ（\ textbf {1}））的$ n $中有一个单一元素$ω_0$ $δ（a）= u_ {w_0^{*}} \ left（φ（p \ circ a） +φ（（（（（\ textbf {1} -p）-p）\ circ a^*）\ right），$ right），所有$ a \ in \ mathfrak {m} $。
Let $M$ and $N$ be unital Jordan-Banach algebras, and let $M^{-1}$ and $N^{-1}$ denote the sets of invertible elements in $M$ and $N$, respectively. Suppose that $\mathfrak{M}\subseteq M^{-1}$ and $\mathfrak{N}\subseteq N^{-1}$ are clopen subsets of $M^{-1}$ and $N^{-1}$, respectively, which are closed for powers, inverses and products of the form $U_{a} (b)$. In this paper we prove that for each surjective isometry $Δ: \mathfrak{M}\to \mathfrak{N}$ there exists a surjective real-linear isometry $T_0: M\to N$ and an element $u_0$ in the McCrimmon radical of $N$ such that $Δ(a) = T_0(a) +u_0$ for all $a\in \mathfrak{M}$.\smallskip Assuming that $M$ and $N$ are unital JB$^*$-algebras we establish that for each surjective isometry $Δ: \mathfrak{M}\to \mathfrak{N}$ the element $Δ(\textbf{1}) =u$ is a unitary element in $N$ and there exist a central projection $p\in M$ and a complex-linear Jordan $^*$-isomorphism $J$ from $M$ onto the $u^*$-homotope $N_{u^*}$ such that $$Δ(a) = J(p\circ a) + J ((\textbf{1}-p) \circ a^*),$$ for all $a\in \mathfrak{M}$. Under the additional hypothesis that there is a unitary element $ω_0$ in $N$ satisfying $U_{ω_0} (Δ(\textbf{1})) = \textbf{1}$, we show the existence of a central projection $p\in M$ and a complex-linear Jordan $^*$-isomorphism $Φ$ from $M$ onto $N$ such that $$Δ(a) = U_{w_0^{*}} \left(Φ(p\circ a) + Φ((\textbf{1}-p) \circ a^*)\right),$$ for all $a\in \mathfrak{M}$.
