Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Spectral permanence for joint spectra
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by Raul E. Curto
Trans. Amer. Math. Soc. 270 (1982), 659-665
DOI: https://doi.org/10.1090/S0002-9947-1982-0645336-8

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 \[ \hat a = \left ( {\begin {array}{*{20}{c}} {{d_1}} & {} & {} \\ {d_2^{\ast }} & {{d_3}} & {} \\ {} & {d_4^{\ast }} & \ddots \\ \end {array} } \right ) \in L\left ( {A \otimes {{\mathbf {C}}^{{2^{n - 1}}}}} \right )\] 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.
References
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Bibliographic Information
  • © Copyright 1982 American Mathematical Society
  • 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