Intrinsic characterization of the space \(c_0(A)\) in the class of Banach lattices (Q1387344)

For a given set \(A\), the author denotes by \(c_0(A)\) the set of all real-number functions such that for any \(\varepsilon> 0\) the set of all \(x\in A\), for which \(| f(x)|\geq\varepsilon\) is finite. The set \(c_0(A)\) can be considered as a Banach lattice with respect to pointwise operation of sup-norm. For an arbitrary vector lattice the following properties are considered: (1) for any net \((x_\alpha)\) and any element \(x\) the condition \(x_\alpha\mathring\to x\) is equivalent to \(\| x_\alpha- x\|\to 0\), (2) for any sequence \((x_n)\) and any element \(x\) the condition \(x_n\mathring\to x\) is equivalent to \(\| x_n- x\|\to 0\). The following theorems are proved: 1. For any Banach lattice the following conditions are equivalent: (a) \(E\) is lineary and lattice isomorphic to \(c_0(A)\) for some set \(A\), (b) \(E\) is of countable type and satisfies property (2). 2. A normed lattice has property (1) if and only if it is finite-dimensional. As a corollary, the author proves: 1. For any Banach lattice \(E\) the following conditions are equivalent: (a) \(E\) is linearly and lattice isomorphic to the Banach space \(c_0\) of all real sequences converging to zero, (b) \(E\) is an infinite-dimensional Banach lattice of countable type with weak unit, in which convergence of the sequences in norm is equivalent to the other convergence. 2. An atom-free Banach lattice with order-continuous norm has no nonzero lattice homomorphisms.