On circular units over the cyclotomic Z_p-Extension of an abelian field

Let K be an abelian number field of conductor f and p a prime. Sinnott defined circular units C_K^s of K using the norm maps from the cyclotomic units of the nth cyclotomic fields for all n and Washington defined circular units C_K^w of K to be the Galois invariant of the cyclotomic units of the f th cyclotomic field. In this note, we investigate a question raised by Kolster in [5] of whether the projective limits of circular units of Sinnott and Washington over the cyclotomic Z _p-extension of K are equal. Belliard [2] and Kučera [6] found independently some counter examples to this question. The purpose of this note is to find some conditions on the ground field K under which Kolster's question has an affirmative answer.