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Subalgebras of modular annihilator algebras

Published online by Cambridge University Press:  24 October 2008

B. A. Barnes
B. A. Barnes
Affiliation:
University of Oregon
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Throughout this paper we deal only with complex and semi-simple algebras. Let B be such an algebra. We denote the socle of B as SB. B is a modular annihilator algebra if B/SB is a radical algebra, i.e. if every element of B is quasi-regular modulo the socle of B; see (1) or (12). Now assume that B is a modular annihilator algebra and a Banach algebra. Then any semi-simple closed subalgebra of B is a modular annihilator algebra by ((4), Cor. to Theorem 4·2,). It is not true, however, that any semi-simple subalgebra A of B is a modular annihilator algebra, even when A is a Banach algebra in some norm. We give a simple example to illustrate this. Let A be the algebra of all complex functions f, continuous on the closed unit disk D in the complex plane, analytic in the interior of D, and such that f(0) = 0. A is a Banach algebra in the usual sup norm over D. Now consider the norm on A defined by

Let B be the completion of A in this norm. A has an involution * defined by and also ‖ff*‖ = ‖f‖2 for all f ∈ A. Therefore B is a B*-algebra. It is not difficult to verify that the only non-zero multiplicative linear functionals on A which are continuous with respect to the norm ‖·‖, are the point evaluations at 1/n, n = 1, 2 …. It follows that every non-zero multiplicative linear functional on B is an extension of one of these point evaluations to B. Thus B can be identified with the algebra of all complex sequences which converge to zero.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 66 , Issue 1 , July 1969 , pp. 5 - 12
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

REFERENCES

(1)Barnes, B. A.Modular annihilator algebras. Canad. J. Math. 18 (1966), 566–578.CrossRefGoogle Scholar
(2)Barnes, B. A.Algebras with the spectral expansion property, Illinois J. Math. 11(1967), 284–290.CrossRefGoogle Scholar
(3)Babnes, B. A. A.generalized Fredholm theory for certain maps in the regular representation of an algebra. Canad. J. Math. 20 (1968), 495–504.Google Scholar
(4)Barnes, B. A.On the existence of minimal ideals in a Banach algebra. Trans. Amer. Math. Soc. 133 (1968), 511–517.CrossRefGoogle Scholar
(5)Dunford, N. and Schwartz, J.Linear operators, Part II (Interscience Publishers, N.Y., 1963).Google Scholar
(6)Grove, L. C.A generalized group algebra for compact groups. Studia Math. 26 (1965), 73–90.CrossRefGoogle Scholar
(7)Jacobson, N.Structure of rings. A.M.S. Colloquium Publications, vol. 37.Google Scholar
(8)Rickart, C. E.Banach algebras (D. Van Nostrand Co. Inc.; Princeton, N. J., 1960).Google Scholar
(9)Wood, G. V.A generalization of the Peter-Weyl theorem. Proc. Cambridge Philos. Soc. 63 (1967), 937–945.CrossRefGoogle Scholar
(10)Yood, B.Homomorphisms on normed algebras, Pacific J. Math. 8 (1958), 373–381.CrossRefGoogle Scholar
(11)Yood, B.Faithful *-representations of normed algebras. Pacific J. Math. 10 (1960), 345–363.CrossRefGoogle Scholar
(12)Yood, B.Ideals in topological rings. Canad. J. Math. 16 (1964), 28–45.CrossRefGoogle Scholar