On cardinal numbers related to the Banach spaces Lp( G) and M( G)

Abstract

Let G be an infinite compact group and Σ its dual object. Let m( G) denote the least cardinal number of an open base of the topology of G. In [3, 28.2] it is proved that m( G)dim L 2( G) and in [3, 28.52] that # G2 m( G) . In this paper it is proved that L p ( G) is isometrically isomorphic to L p ( T m( G) ) (1 ⩽ p ⩽ ∞). It is proved that when G and H are compact groups with M( G) isometrically isomorphic to M( H), then L 1( G) is isometrically isomorphic to L 1( H). Further it is proved that M( G) is isometrically isomorphic to M( T m( G) ) in case G is abelian.