A characterization of the group algebras of the finite groups

Marc A. Rieffel

Vol. 16 (1966), No. 2, 347–363

DOI: 10.2140/pjm.1966.16.347

Abstract

The following is proved:

MAIN THEOREM. Let A be a finite dimensional Archemedian lattice ordered algebra which satisfies the following axioms:

MO If f,g,h ∈ A, and if f ≧ 0, then

1. f ∗ (g ∨ h) = ∨{f₁ ∗ g + f₂ ∗ h : f₁ ≧ 0,f₂ ≧ 0,f₁ + f₂ = f}.
2. (g ∨ h) ∗ f = ∨{g ∗ f₁ + h ∗ f₂ : f₁ ≧ 0,f₂ ≧ 0,f₁ + f₂ = f}.
3. If f,g ∈ A, and if f > 0, g > 0, then f ∗ g > 0.

Then there exists a finite group G such that A is order and algebra isomorphic to the group algebra of G.

Some similar results are obtained for finite semigroups, and a few applications of these results are given. In particular it is shown that the second cohomology group, H²(S,R), of any finite commutative semigroup, S, with coefficients in the additive group of real numbers, R, is trivial.

Mathematical Subject Classification

Primary: 20.80

Milestones

Received: 5 August 1964

Revised: 13 October 1964

Published: 1 February 1966