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), 189-198

MSC (1991):
Primary 46L05, 47D30

MathSciNet review:
1346968

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Abstract | References | Similar Articles | Additional Information

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 .

**[Cu1]**Raul E. Curto,*Fredholm and invertible 𝑛-tuples of operators. The deformation problem*, Trans. Amer. Math. Soc.**266**(1981), no. 1, 129–159. MR**613789**, 10.1090/S0002-9947-1981-0613789-6**[Cu2]**Raul E. Curto,*Spectral permanence for joint spectra*, Trans. Amer. Math. Soc.**270**(1982), no. 2, 659–665. MR**645336**, 10.1090/S0002-9947-1982-0645336-8**[DLW]**J. Daughtry, A. Lambert, and B. Weinstock,*Operators on -algebras induced by conditional expectations*, Rocky Mountain Journal of Mathematics**25**(1995), 1243-1275.**[LW]**A. Lambert and B. Weinstock,*A class of operator algebras induced by probabilistic conditional expectations*, Michigan Math. J.**40**(1993), 359-376. CMP**93:14****[Pas]**William L. Paschke,*Inner product modules over 𝐵*-algebras*, Trans. Amer. Math. Soc.**182**(1973), 443–468. MR**0355613**, 10.1090/S0002-9947-1973-0355613-0**[R]**C. E. Rickart,*On spectral permanence for certain Banach algebras*, Proc. Amer. Math. Soc.**4**(1953), 191–196. MR**0052700**, 10.1090/S0002-9939-1953-0052700-4**[W-O]**N. E. Wegge-Olsen,*𝐾-theory and 𝐶*-algebras*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. A friendly approach. MR**1222415**

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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/S0002-9939-97-03536-3

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