The Mackey-Gleason-Bunce-Wright problem for vector-valued measures on projections in a JBW$^*$-algebra
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| Format: | Preprint |
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2025
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| _version_ | 1866908517595611136 |
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| author | Escolano, Gerardo M. Peralta, Antonio M. Villena, Armando R. |
| author_facet | Escolano, Gerardo M. Peralta, Antonio M. Villena, Armando R. |
| contents | Let $\mathcal{P} (\mathfrak{J})$ denote the lattice of projections of a JBW$^*$-algebra $\mathfrak{J}$, and let $X$ be a Banach space. A bounded finitely additive $X$-valued measure on $\mathcal{P}(\mathfrak{J})$ is a mapping $μ: \mathcal{P}(\mathfrak{J}) \rightarrow X$ satisfying:
$(a)$ $μ(p +q) = μ(p) + μ(q)$, whenever $p \circ q = 0$ in $\mathcal{P} (\mathfrak{J})$,
$(b)$ $\sup \{ \| μ(p)\| \, : \, p \in \mathcal{P} (\mathfrak{J})\} < \infty$.
In this paper we establish a Mackey-Gleason-Bunce-Wright theorem by showing that if $\mathfrak{J}$ contains no type $I_2$ direct summand, every bounded finitely additive measure $μ: \mathcal{P}(\mathfrak{J}) \rightarrow X$ admits an extension to a bounded linear operator from $\mathfrak{J}$ to $X$. This solves a long-standing open conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03213 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Mackey-Gleason-Bunce-Wright problem for vector-valued measures on projections in a JBW$^*$-algebra Escolano, Gerardo M. Peralta, Antonio M. Villena, Armando R. Operator Algebras Functional Analysis Let $\mathcal{P} (\mathfrak{J})$ denote the lattice of projections of a JBW$^*$-algebra $\mathfrak{J}$, and let $X$ be a Banach space. A bounded finitely additive $X$-valued measure on $\mathcal{P}(\mathfrak{J})$ is a mapping $μ: \mathcal{P}(\mathfrak{J}) \rightarrow X$ satisfying: $(a)$ $μ(p +q) = μ(p) + μ(q)$, whenever $p \circ q = 0$ in $\mathcal{P} (\mathfrak{J})$, $(b)$ $\sup \{ \| μ(p)\| \, : \, p \in \mathcal{P} (\mathfrak{J})\} < \infty$. In this paper we establish a Mackey-Gleason-Bunce-Wright theorem by showing that if $\mathfrak{J}$ contains no type $I_2$ direct summand, every bounded finitely additive measure $μ: \mathcal{P}(\mathfrak{J}) \rightarrow X$ admits an extension to a bounded linear operator from $\mathfrak{J}$ to $X$. This solves a long-standing open conjecture. |
| title | The Mackey-Gleason-Bunce-Wright problem for vector-valued measures on projections in a JBW$^*$-algebra |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2509.03213 |