Uncountable families of prime z-ideals in C_o (R)

摘要

Denote by c = 250 the cardinal of continuum. We construct an intriguing family (P_a:a _ c) of prime z-ideals in Co (R,) with the following properties: (i) if f __ Р_io for some io E c, then f i_ P_i for all but finitely many ii_c; (ii)i__ii_i_0 Р_ii_P_io for each i_o i_c. As an application, assuming the continuum hypothesis, there exists a homomorphism from C_o (R,) into a Banach algebra whose continuity ideal is not the intersection of any countable family of prime ideals in C_o(R,). We also construct a well-ordered increasing chain, as well as a well-ordered decreasing chain, of order type K of prime z-ideals in C_o(R,) for any ordinal K of cardinality c.