Spectral permanence for joint spectra

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.