Spectral permanence - Imported Knowlpedia Content

Statement

Let $B\subseteq A$ be a $C^*$ -subalgebra. If $A$ and $B$ are unital and share the same identity, then every $b\in B$ has

\sigma_B(b)=\sigma_A(b),

where the spectra are those of Banach-algebra elements. Equivalently, whenever $b$ is invertible in $A$ , its inverse already lies in $B$ . For nonunital inclusions, the same assertion holds after passing to the compatible unitizations. This theorem is spectral permanence. Its unit convention is essential: viewing a nonunital subalgebra as though it shared the ambient unit can change the status of zero in the spectrum.

Proof idea

For $b\in B$ , invertibility of $b$ in $A$ implies that $b^*b$ is bounded below by a positive scalar. Continuous functional calculus inside the $C^*$ -algebra generated by $b^*b$ then produces $(b^*b)^{-1}$ in $B$ , and

b^{-1}=(b^*b)^{-1}b^*.

Applying the same reasoning to shifts $b-\lambda1$ gives equality of spectra Murphy, §2.1.

Consequences

Positivity, spectral radius, and continuous functional calculus for an element of $B$ may be computed intrinsically in $B$ or in the ambient algebra $A$ . In particular, if $b\in B$ is normal and $f$ is continuous on its spectrum, then $f(b)\in B$ . This is one reason norm-closed $*$ -subalgebras are the natural subobjects in $C^*$ -algebra theory.

Conventions and scope

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §2.1 on $C^*$ -subalgebras and spectral permanence.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.2 on spectra and functional calculus.