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

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
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