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Stability of characters and filters for weighted semilattices. (English) Zbl 07310708
Summary: We continue the study of the AMNM property for weighted semilattices that was initiated in [Y. Choi, J. Aust. Math. Soc. 95, No. 1, 36–67 (2013; Zbl 1317.46033)]. We reformulate this in terms of stability of filters with respect to a given weight function, and then provide a combinatorial condition which is necessary and sufficient for this “filter stability” property to hold. Examples are given to show that this new condition allows for easier and unified proofs of some results in loc. cit., and furthermore allows us to verify the AMNM property in situations not covered by the results of that paper. As a final application, we show that for a large class of semilattices, arising naturally as union-closed set systems, one can always construct weights for which the AMNM property fails.

MSC:
46J10 Banach algebras of continuous functions, function algebras
06B05 Structure theory of lattices
43A10 Measure algebras on groups, semigroups, etc.
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