Remarkable classes of unital AM-spaces. (English) Zbl 0792.46004

Abramovich, Y. A.; Wickstead, A. W.

J. Math. Anal. Appl. 180, No. 2, 398-411 (1993).

Summary: We define and investigate two classes of unital Banach AM-spaces, the elements of which are the sums of continuous functions and discrete functions. Neither class is almost Dedekind \(\sigma\)-complete, although one has the Cantor property. One class has the rather rare property of having a sequentially order continuous norm and we deduce that any \(C(K)\) space can be embedded as a sublattice of a \(C(X)\) space with a sequentially order continuous norm. Finally we identify the order continuous and sequentially order continuous duals of spaces in these classes, which promise to be a rich source of further examples.

MSC:

46A40

Ordered topological linear spaces, vector lattices

Keywords:

unital Banach AM-spaces; almost Dedekind \(\sigma\)-complete; Cantor property