Vol. 31, No. 3, 1969

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ISSN: 0030-8730
Representation of L-groups and F-rings

John Dauns

Vol. 31 (1969), No. 3, 629–654
Abstract

Consider an f-algebra A with identity (i.e., a,b,c A, αΛb = 0, c 0 cab = acΛb = 0) over the rationals Q. Let be the maximal l-ideals of A.

THEOREM I. If 1 a A 1∕a A, then each A∕M,M ∈ℳ is a totally ordered division ring, and A can be embedded into a real f-algebra.

THEOREM II. A fiber-bundle or sheaf-like structure

π : E ≡ ∪{A∕M |M ∈ ℳ } → ℳ, π−1(M ) = A∕M ⊂ E

is constructed. Assume ∩ℳ = {0}; has the hull-kernel topology. All continuous cross sections σ;ℳ→ E(π σ = identity) form a partial algebra Γ(,E) containing an isomorphic copy of AÂ Γ(,E). Let A = {a A||a| < n1 some integer n}. If

  1. 1 a A1∕α A then A Γ(,E), where  is order dense in Γ(,E). If in addition
  2. A is complete with respect to the absolute value |a|,a A,

then AÂ = Γ(,E).

Mathematical Subject Classification
Primary: 06.90
Milestones
Received: 12 September 1967
Revised: 27 January 1969
Published: 1 December 1969
Authors
John Dauns