Vol. 16, No. 2, 1966

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ISSN: 0030-8730
A characterization of the group algebras of the finite groups

Marc A. Rieffel

Vol. 16 (1966), No. 2, 347–363
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
Authors
Marc A. Rieffel
University of California, Berkeley
Berkeley CA
United States