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Simplex algebras and their representation

Published online by Cambridge University Press:  18 May 2009

M. S. Vijayakumar
M. S. Vijayakumar
Affiliation:
Dalhousie University, Halifax, N.S., Canada
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This paper establishes a relationship (Theorem 4.1) between the approaches of A. C. Thompson [8, 9] and E. G. Effros [2] to the representation of simplex algebras, that is, real unital Banach algebras that are simplex spaces with the unit for order identity. It proves that the (nonempty) interior of the associated cone is contained in the principal component of the set of all regular elements of the algebra. It also conjectures that each maximal ideal (in the order sense—see below) of a simplex algebra contains a maximal left ideal of the algebra. This conjecture and other aspects of the relationship are illustrated by considering algebras of n × n real matrices.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 14 , Issue 2 , September 1973 , pp. 136 - 144
Copyright
Copyright © Glasgow Mathematical Journal Trust 1973

References

REFERENCES

1.Bohnenblust, H. F. and Karlin, S., Geometrical properties of the unit sphere of Banach algebras, Ann. of Math. 62 (1955), 217–229.CrossRefGoogle Scholar
2.Effros, E. G., Structure in simplexes, Acta Math. 117 (1967), 103–121.CrossRefGoogle Scholar
3.Ingelstam, L., A vertex property for Banach algebras with identity, Math. Scand. 11 (1962), 22–32.CrossRefGoogle Scholar
4.Jameson, Graham, Ordered linear spaces, Lecture Notes in Mathematics (141) (Springer Verlag, New York, 1970).CrossRefGoogle Scholar
5.Klee, V. L., Separation properties for convex cones, Proc. Amer. Math. Soc. 6 (1955), 313–318.CrossRefGoogle Scholar
6.Kelley, J. L. and Namioka, I., Linear topological spaces (New York, 1963).CrossRefGoogle Scholar
7.Ng, Kung Fu, A representation theorem for partially ordered Banach algebras, Proc. Cambridge Philos. Soc. 64 (1968), 53–59.CrossRefGoogle Scholar
8.Thompson, A. C., On the geometry of Banach algebras, Notices Amer. Math. Soc. 16 (1969), 274.Google Scholar
9.Thompson, A. C. and Vijayakumar, M. S., An order-preserving representation theorem for complex Banach algebras and some examples, Glasgow Math. J. 14 (1973), 128–135.CrossRefGoogle Scholar