Vol. 17, No. 3, 1966

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On the multiplicative extension property

Richard Albert Cleveland and Sandra Cleveland

Vol. 17 (1966), No. 3, 413–421

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 Mc(Q) has the m.e.p., then (i) for every nonzero t G, the set Q (Q t) has μ-measure 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 t0, Q (Q t) is countably infinite, and Mc(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.

Mathematical Subject Classification
Primary: 46.50
Received: 8 March 1965
Published: 1 June 1966
Richard Albert Cleveland
Sandra Cleveland