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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Invariance of spectrum for representations
of $\mathbf C^*$-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: Let $\mathcal K $ be a Banach space, $\mathcal B $ a unital $\mathrm C^*$-algebra, and $\pi :\mathcal B \to\mathcal L (\mathcal K )$ an injective, unital homomorphism. Suppose that there exists a function $\gamma :\mathcal K \times \mathcal K\to \mathbb R^+$ such that, for all $k,k_1,k_2\in \mathcal K$, and all $b\in \mathcal B$,

(a) $\gamma (k,k)=\|k\|^2$,

(b) $\gamma (k_1,k_2)\le \|k_1\|\,\|k_2\|$,

(c) $\gamma (\pi _bk_1,k_2)=\gamma (k_1,\pi _{b^*}k_2)$.
Then for all $b\in \mathcal B$, the spectrum of $b$ in $\mathcal B $ equals the spectrum of $\pi _b$ as a bounded linear operator on $\mathcal K $. If $\gamma $ satisfies an additional requirement and $\mathcal B $ is a $\mathrm W ^*$-algebra, then the Taylor spectrum of a commuting $n$-tuple $b=(b_1,\dotsc ,b_n)$ of elements of $\mathcal B $ equals the Taylor spectrum of the $n$-tuple $\pi _b$ in the algebra of bounded operators on $\mathcal K $. Special cases of these results are (i) if $\mathcal K $ is a closed subspace of a unital $\mathrm C^*$-algebra which contains $\mathcal B $ as a unital $\mathrm C ^*$-subalgebra such that $\mathcal {BK}\subseteq \mathcal K$, and $b\mathcal K =\{0\}$ only if $b=0$, then for each $b\in \mathcal B$, the spectrum of $b$ in $\mathcal B $ is the same as the spectrum of left multiplication by $b$ on $\mathcal K $; (ii) if $\mathcal A $ is a unital $\mathrm C ^*$-algebra and $\mathcal J $ is an essential closed left ideal in $\mathcal A $, then an element $a$ of $\mathcal A $ is invertible if and only if left multiplication by $a$ on $\mathcal J $ is bijective; and (iii) if $\mathcal A $ is a $\mathrm C ^* $-algebra, $\mathcal E $ is a Hilbert $\mathcal A $-module, and $T$ is an adjointable module map on $\mathcal E $, then the spectrum of $T$ in the $\mathrm C ^*$-algebra of adjointable operators on $\mathcal E $ is the same as the spectrum of $T$ as a bounded operator on $\mathcal E $. If the algebra of adjointable operators on $\mathcal E $ is a $\mathrm W ^*$-algebra, then the Taylor spectrum of a commuting $n$-tuple of adjointable operators on $\mathcal E $ is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on $\mathcal E $.


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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
PII: S 0002-9939(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