Vol. 53, No. 2, 1974

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Functional representation of algebraic intervals

Robert Edward Jamison, II

Vol. 53 (1974), No. 2, 399–423

Motivated by some examples from the study of axiomatic convexity, we define a class of objects (in real algebras with 1) whose algebraic properties mimic those of the unit interval. These objects, called intervals, have quite a bit of structure in themselves. In particular, in a Banach algebra a compact interval must be finite dimensional. Even more striking is the main result which shows that any interval satisfying a very modest boundedness condition is commutative and can be represented by continuous functions from a compact Hausdorff space into the unit interval. This leads to a number of corollaries in analysis and topology.

Mathematical Subject Classification 2000
Primary: 46H05
Secondary: 46E15
Received: 5 July 1972
Revised: 28 January 1974
Published: 1 August 1974
Robert Edward Jamison, II