Homomorphisms of group algebras with squared norm less than

Nigel Kalton and G. V. Wood

Vol. 62 (1976), No. 2, 439–460

DOI: 10.2140/pjm.1976.62.439

Abstract

We show that two locally compact abelian groups G₁ and G₂ are isomorphic if there exists an aigebra isomorphism T of L¹(G₁) onto L¹(G₂) with ∥T∥ < . This constant is best possible. The same result is proved for locally compact connected groups, but for the general locally compact group, the result is proved under the hypothesis ∥T∥ < 1.246. Similar results are given for the algebras C(G) and L^∞(G) when G is compact. In the abelian case, we give a representation theorem for isomorphisms satisfying ∥T∥ < .

Mathematical Subject Classification 2000

Primary: 43A20

Milestones

Received: 30 June 1975

Published: 1 February 1976