Restrictions of Convex Subsets of C(X)

Bounded extension property and 𝑝-sets

“…Then It is well known that in the case when A is the disc algebra and F Ç T, the unit circle, then F is a /»-set for A if and only if F has one-dimensional Lebesgue measure 0. Hence Theorem 2.1 can be considered as a generalization of Proposition 4.3 of [6]. Thus Theorem 2.1 answers the second question of §5 of [6], affirmatively.…”

“…Hence Theorem 2.1 can be considered as a generalization of Proposition 4.3 of [6]. Thus Theorem 2.1 answers the second question of §5 of [6], affirmatively.…”

“…F if and only if F is ap-set for A. We also answer two questions raised in [6] using this result, although our earlier arguments could be used. The first result deals with elements in an arbitrary uniform algebra with nonnegative real parts, while the second result concerns the disc algebra.…”

“…Consequences. In [6] the following result is proved. (b) Given a pair f G S, g G S\F and a lower semicontinuous function \¡/\ X ->(0,oo] such that \fix) -g(x)\ < ¡p(x) for all x G F, then there exists an extension g £ S of g such that \f(x) -g(x)\ < \p(x)for all x £ X.…”

“…The main result. The proof of the following theorem is the same as the proof of Proposition 4.3 of [6]. Since we are dealing with a more general situation, we give the argument for the sake of completeness.…”