Uniqueness of norm on L-p(G) and C(G) when G is a compact group

摘要

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L-0(2) (G) and X is either L-p(G) for 1 ≤ ∞ or C(G), then we show that every complete norm | (.) | on X that makes translations from (X, | (.) |) into itself continuous is equivalent to ∥ (.) ∥(p) or ∥ (.) ∥(∞) respectively. If 1less than or equal toinfinity and every left invariant linear functional on L-p(G) is a constant multiple of the Haar integral, then we show that every complete norm (.) on L-p(G) that makes translations from (L-p(G), (.) ) into itself continuous and that makes the map t-->L-t from G into L(L-p(G), (.) bounded is equivalent to parallel to (.) parallel to(p). (C) 2002 Elsevier Science (USA). All rights reserved. [References: 16]