Quadratic & additive mappings on operator commuting elements in JBW*-algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912971141152768 |
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| author | Escolano, Gerardo M. Hamhalter, Jan Peralta, Antonio M. Villena, Armando R. |
| author_facet | Escolano, Gerardo M. Hamhalter, Jan Peralta, Antonio M. Villena, Armando R. |
| contents | Let $\mathfrak{A}$ and $\mathfrak{B}$ be JBW$^*$-algebras whose sets of unitaries are denoted by $\mathcal{U}(\mathfrak{A})$ and $\mathcal{U}(\mathfrak{B})$, respectively. We show that $\mathcal{U}(\mathfrak{A})$ is closed for Jordan products of operator commuting pairs inside itself. Assuming that $\mathfrak{A}$ and $\mathfrak{B}$ are JBW$^*$-algebras without direct summands of type $I_1$ or $I_2$, we prove that for each bicontinuous bijection $Φ: \mathcal{U}(\mathfrak{A}) \rightarrow \mathcal{U}(\mathfrak{B})$ satisfying $Φ(u \circ v) = Φ(u)\circ Φ(v),$ whenever $u$ and $v$ are operator commuting unitaries in $\mathfrak{A}$, there exist a linear Jordan $^*$-isomorphism $θ: \mathfrak{A} \rightarrow \mathfrak{B}$, a real linear mapping $β: \mathfrak{A_{sa}}\rightarrow Z(\mathfrak{B}_{sa})$, and an invertible central element $c \in \mathfrak{B}_{sa}$ such that $$ Φ(e^{i a}) = e^{i β(a)}\circ e^{i c\circθ(a)} = e^{i β(a)} \circ θ\left( e^{i θ^{-1}( c )\circ a}\right),$$ for all $a\in \mathfrak{A}_{sa}$. The conclusion improves when $\mathfrak{A}$ is a JBW$^*$-algebra factor not of type $I_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16687 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quadratic & additive mappings on operator commuting elements in JBW*-algebras Escolano, Gerardo M. Hamhalter, Jan Peralta, Antonio M. Villena, Armando R. Operator Algebras Functional Analysis Let $\mathfrak{A}$ and $\mathfrak{B}$ be JBW$^*$-algebras whose sets of unitaries are denoted by $\mathcal{U}(\mathfrak{A})$ and $\mathcal{U}(\mathfrak{B})$, respectively. We show that $\mathcal{U}(\mathfrak{A})$ is closed for Jordan products of operator commuting pairs inside itself. Assuming that $\mathfrak{A}$ and $\mathfrak{B}$ are JBW$^*$-algebras without direct summands of type $I_1$ or $I_2$, we prove that for each bicontinuous bijection $Φ: \mathcal{U}(\mathfrak{A}) \rightarrow \mathcal{U}(\mathfrak{B})$ satisfying $Φ(u \circ v) = Φ(u)\circ Φ(v),$ whenever $u$ and $v$ are operator commuting unitaries in $\mathfrak{A}$, there exist a linear Jordan $^*$-isomorphism $θ: \mathfrak{A} \rightarrow \mathfrak{B}$, a real linear mapping $β: \mathfrak{A_{sa}}\rightarrow Z(\mathfrak{B}_{sa})$, and an invertible central element $c \in \mathfrak{B}_{sa}$ such that $$ Φ(e^{i a}) = e^{i β(a)}\circ e^{i c\circθ(a)} = e^{i β(a)} \circ θ\left( e^{i θ^{-1}( c )\circ a}\right),$$ for all $a\in \mathfrak{A}_{sa}$. The conclusion improves when $\mathfrak{A}$ is a JBW$^*$-algebra factor not of type $I_2$. |
| title | Quadratic & additive mappings on operator commuting elements in JBW*-algebras |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2603.16687 |