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Dense Q-subalgebras of Banach and C*-algebras and unbounded derivations of Banach and C*-algebras

Published online by Cambridge University Press:  20 January 2009

E. Kissin  and
V. S. Shulman
E. Kissin
Affiliation:
School of Mathematical Sciences, University of North London, Holloway, London N7 8DB, Great Britain
V. S. Shulman
Affiliation:
Department of Mathematics, Polytechnic Institute of Vologda, Vologda, USSR
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Abstract

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The paper studies dense Q-subalgebras of Banach and C*-algebras. It proves that the domain D(δ) of a closed unbounded derivation δ of a Banach unital algebra A automatically contains the identity and is a Q-subalgebra of A, so that SpA(x) = SpD(δ)(x) for all x∈D(δ). The paper shows that every finite-dimensional semisimple representation of a Q-subalgebra is continuous. It also shows that if π is an injective *-homomorphism of a dense locally normal Q*-subalgebra B of a C*-algebra, then ‖x‖≦‖π(x)‖ for all x∈B. The paper studies the link between closed ideals of a Banach algebra A and of its dense subalgebra B. In particular, if A is a C*-algebra and B is a locally normal *-subalgebra of A, then I→I∩B is a one-to-one mapping of the set of all closed two-sided ideals in A onto the set of all closed two-sided ideals in B and .

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 2 , June 1993 , pp. 261 - 276
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

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