Invariance of spectrum for representations of algebras on Banach spaces
Authors:
John Daughtry, Alan Lambert and Barnet Weinstock
Journal:
Proc. Amer. Math. Soc. 125 (1997), 189198
MSC (1991):
Primary 46L05, 47D30
MathSciNet review:
1346968
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Abstract: Let be a Banach space, a unital algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all , (a) , (b) , (c) . Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a algebra, then the Taylor spectrum of a commuting tuple of elements of equals the Taylor spectrum of the tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital algebra which contains as a unital subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a algebra, is a Hilbert module, and is an adjointable module map on , then the spectrum of in the algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a algebra, then the Taylor spectrum of a commuting tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .
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Additional Information
John Daughtry
Affiliation:
Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
Email:
madaught@ecuvm.cis.ecu.edu
Alan Lambert
Email:
fma00all@unccvm.uncc.edu
Barnet Weinstock
Affiliation:
Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223
Email:
fma00bmw@unccvm.uncc.edu
DOI:
http://dx.doi.org/10.1090/S0002993997035363
PII:
S 00029939(97)035363
Received by editor(s):
July 14, 1995
Additional Notes:
The second and third authors wish to thank David Larson and the Department of Mathematics at Texas A & M University for the opportunity to attend the 1994 Summer Workshop in Probability and Linear Analysis where some of the ideas in this paper were developed.
The work of the second and third authors was partially supported by Faculty Research Grants from the University of North Carolina at Charlotte.
Communicated by:
Palle E. T. Jorgensen
Article copyright:
© Copyright 1997 American Mathematical Society
