A subspace M of a Banach
algebra B is said to have the multiplicative extension property (abbr. m.e.p.) if
whenever L is a linear functional on M of norm not greater than one, L is the
restriction to M of a multiplicative linear functional on B. This property is
considered in two settings—the measure algebra M(G) of a suitable group, and the
disc algebra A(D) of functions analytic in the unit disc with continuous boundary
values. The following theorems are proved.
Theorem 2. If Q is a compact subset of G such that M_{c}(Q) has the m.e.p., then
(i) for every nonzero t ∈ G, the set Q ∩ (Q − t) has μmeasure zero for every
continuous measure μ on G, and (ii) m(Q) = 0, where m is the Haar measure for
G.
Theorem 3. Suppose G contains an independent Cantor set. Then there exists a
compact subset Q of G such that for infinitely many t≠0, Q ∩ (Q − t) is countably
infinite, and M_{c}(Q) has the m.e.p.
Theorem 4. There exist infinite dimensional subspaces of A(D) with the
m.e.p.
These last two theorems are proved by constructing examples using a special
decomposition of the Cantor set. This decomposition is presented in a separate
section to simplify notation.
