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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Spectral permanence for joint spectra


Author: Raul E. Curto
Journal: Trans. Amer. Math. Soc. 270 (1982), 659-665
MSC: Primary 46L05; Secondary 47A10
DOI: https://doi.org/10.1090/S0002-9947-1982-0645336-8
MathSciNet review: 645336
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Abstract: For a $ {C^{\ast}}$-subalgebra $ A$ of a $ {C^{\ast}}$-algebra $ B$ and a commuting $ n$-tuple $ a = ({a_1}, \ldots ,{a_n})$ of elements of $ A$, we prove that $ \operatorname{Sp} (a,\,A) = \operatorname{Sp} (a,\,B)$, where $ \operatorname{Sp} $ denotes Taylor spectrum. As a consequence we prove that $ 0 \notin \operatorname{Sp} (a,\,A)$ if and only if

$\displaystyle \hat a = \left( {\begin{array}{*{20}{c}} {{d_1}} & {} & {} \\ {d_... ...d{array} } \right) \in L\left( {A \otimes {{\mathbf{C}}^{{2^{n - 1}}}}} \right)$

is invertible, where $ {d_i}$ is the $ i$th boundary map in the Koszul complex for $ A$. More generally, we show that $ {\sigma _{\delta ,k}}(a,\,A) = {\sigma _{\delta ,k}}\left( {a,\,B} \right)$ and $ {\sigma _{\pi ,k}}(a,\,A) = {\sigma _{\pi ,k}}(a,\,B)$ (all $ k$), where $ {\sigma _{\delta ,\cdot}}$ and $ {\sigma _{\pi ,\cdot}}$ are the joint spectra considered by Z. Słodkowski.

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DOI: https://doi.org/10.1090/S0002-9947-1982-0645336-8
Keywords: Spectral permanence, joint spectra, von Neumann algebras, universal representation
Article copyright: © Copyright 1982 American Mathematical Society