Trace algebras
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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.References
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- Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 165 (1972), 389-423
- MSC: Primary 15A03
- DOI: https://doi.org/10.1090/S0002-9947-1972-0294359-3
- MathSciNet review: 0294359