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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Trace algebras

Author: R. P. Sheets
Journal: Trans. Amer. Math. Soc. 165 (1972), 389-423
MSC: Primary 15A03
MathSciNet review: 0294359
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Abstract: We give an algebraic unification for those mathematical structures which possess the abstract properties of finite-dimensional vector spaces: scalars, duality theories, trace functions, etc. The unifying concept is the ``trace algebra,'' which is a set with a ternary operation which satisfies certain generalized associativity and identity laws. Every trace algebra induces naturally an object which (even though no additive structure may be available) possesses a summation operator and inner product which obey the Fourier expansion and other familiar properties. We construct the induced object in great detail. The ultimate results of the paper are: a theorem which shows that the induced object of a ``well-behaved'' trace algebra determines it uniquely; and a theorem which shows that well-behaved trace algebras look, formally, like the trace algebras associated with finite-dimensional vector spaces.

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Keywords: Ternary operation, partial associativity, trace function, inner product, summation operator, bihomogeneous binary maps, duality theory, Riesz representation theorem, Parseval's identity, Fourier expansion
Article copyright: © Copyright 1972 American Mathematical Society