Invariance of spectrum for representations of $\mathrm {C}*$-algebras on Banach spaces
HTML articles powered by AMS MathViewer
- by John Daughtry, Alan Lambert and Barnet Weinstock PDF
- Proc. Amer. Math. Soc. 125 (1997), 189-198 Request permission
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 $\mathcal {E}$.
References
- 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, DOI 10.1090/S0002-9947-1981-0613789-6
- Raul E. Curto, Spectral permanence for joint spectra, Trans. Amer. Math. Soc. 270 (1982), no. 2, 659–665. MR 645336, DOI 10.1090/S0002-9947-1982-0645336-8
- J. Daughtry, A. Lambert, and B. Weinstock, Operators on $C^*$-algebras induced by conditional expectations, Rocky Mountain Journal of Mathematics 25 (1995), 1243–1275.
- A. Lambert and B. Weinstock, A class of operator algebras induced by probabilistic conditional expectations, Michigan Math. J. 40 (1993), 359–376.
- William L. Paschke, Inner product modules over $B^{\ast }$-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. MR 355613, DOI 10.1090/S0002-9947-1973-0355613-0
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- 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
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
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 189-198
- MSC (1991): Primary 46L05, 47D30
- DOI: https://doi.org/10.1090/S0002-9939-97-03536-3
- MathSciNet review: 1346968