论文标题
一个人可以通过由其一组人确定的度量空间来识别两个Unital JB $^*$ - 代数吗？
Can one identify two unital JB$^*$-algebras by the metric spaces determined by their sets of unitaries?
论文作者
论文摘要
让$ m $和$ n $为两个Unital JB $^*$ - 代数，让$ \ Mathcal {u}（m）$和$ \ Mathcal {u}（n）$分别用$ M $和$ n $表示所有一级的集合。我们证明以下陈述是等效的： $（a）$ $ m $，$ n $是（复杂）BANACH空间的同构同构； $（b）$ $ m $，$ n $是同构的同构，作为真正的Banach空间； $（c）$存在溢流等轴测$δ：\ Mathcal {u}（m）\ to \ Mathcal {u}（n）（n）。$ 我们实际上建立了一个更一般的陈述，断言在某些额外的额外条件下，对于每个冲销等轴测$δ：\ Mathcal {u}（m）\ to \ nathcal {u}（n）$，我们可以找到一个与N $相关的N $ comcy $Δ$ fumentive realinearsememetry $ c $ ex $ e} $ e}如果我们假设$ m $和$ n $是jbw $^*$ - 代数，则每个溢流等轴测$δ：\ mathcal {u}（m）\ to \ to \ nathcal {u}（u}（n）$ aff（unique）向$ m $ $ n $ n $ n $ n $ n $ n $ n $ n $ n $。这是hatori-moln {Á} r定理的扩展，以jb $^*$ - 代数的设置。
Let $M$ and $N$ be two unital JB$^*$-algebras and let $\mathcal{U} (M)$ and $\mathcal{U} (N)$ denote the sets of all unitaries in $M$ and $N$, respectively. We prove that the following statements are equivalent: $(a)$ $M$ and $N$ are isometrically isomorphic as (complex) Banach spaces; $(b)$ $M$ and $N$ are isometrically isomorphic as real Banach spaces; $(c)$ There exists a surjective isometry $Δ: \mathcal{U}(M)\to \mathcal{U}(N).$ We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry $Δ:\mathcal{U} (M) \to \mathcal{U} (N)$ we can find a surjective real linear isometry $Ψ:M\to N$ which coincides with $Δ$ on the subset $e^{i M_{sa}}$. If we assume that $M$ and $N$ are JBW$^*$-algebras, then every surjective isometry $Δ:\mathcal{U} (M) \to \mathcal{U} (N)$ admits a (unique) extension to a surjective real linear isometry from $M$ onto $N$. This is an extension of the Hatori--Moln{á}r theorem to the setting of JB$^*$-algebras.
