Geodesic
The positive cone of a Banach lattice. Coincidence of topologies and metrizability
Commentationes Mathematicae Universitatis Carolinae, Tome 64 (2023) no. 4, pp. 475-483
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Let $X$ be a Banach lattice, and denote by $X_+$ its positive cone. The weak topology on $X_+$ is metrizable if and only if it coincides with the strong topology if and only if $X$ is Banach-lattice isomorphic to $l^1(\Gamma)$ for a set $\Gamma$. The weak$^*$ topology on $X_+^*$ is metrizable if and only if $X$ is Banach-lattice isomorphic to a $C(K)$-space, where $K$ is a metrizable compact space.
Let $X$ be a Banach lattice, and denote by $X_+$ its positive cone. The weak topology on $X_+$ is metrizable if and only if it coincides with the strong topology if and only if $X$ is Banach-lattice isomorphic to $l^1(\Gamma)$ for a set $\Gamma$. The weak$^*$ topology on $X_+^*$ is metrizable if and only if $X$ is Banach-lattice isomorphic to a $C(K)$-space, where $K$ is a metrizable compact space.
DOI : 10.14712/1213-7243.2024.004
Classification : 46B42, 46E05, 54E35
Keywords: normed lattice; Banach lattice; positive cone; AM-space; AL-space; Banach lattice $C(K)$; Banach lattice $l^1(\Gamma)$; strong topology; weak topology; weak$^*$ topology; coincidence of topologies; metrizability; nonatomic measure 
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     title = {The positive cone of a {Banach} lattice. {Coincidence} of topologies and metrizability},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
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Lipecki, Zbigniew. The positive cone of a Banach lattice. Coincidence of topologies and metrizability. Commentationes Mathematicae Universitatis Carolinae, Tome 64 (2023) no. 4, pp. 475-483. doi: 10.14712/1213-7243.2024.004

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