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CONVEX IDEALS IN ORDERED GROUP ALGEBRAS AND THE UNIQUENESS OF THE HAAR MEASURE

K. E. AUBERT
Mathematica Scandinavica
Vol. 6, No. 2 (May 30, 1959), pp. 181-188
Published by: Mathematica Scandinavica
Stable URL: http://www.jstor.org/stable/24490196
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
CONVEX IDEALS IN ORDERED GROUP ALGEBRAS AND THE UNIQUENESS OF THE HAAR MEASURE
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Abstract

Let LR1 denote the pointwise ordered convolution ring of all integrable real-valued functions on a locally compact abelian group. The present paper contains proofs of the following results: 1° Let μ be an order-preserving ring homomorphism of LR1 onto an ordered field F. Then the field F is isomorphic to the field of real numbers and μ is the Haar measure of G. 2° An ideal in LR1 which is equal to the intersection of a non-void family of regular maximal ideals in LR1 is convex if and only if it is contained in the ideal 𝔪R0 consisting of functions with zero integral. 3° LR1 can never (i.e. for any G ≠ {e}) be embedded as an ordered ring in a direct product of totally ordered rings.

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