Suppose 𝒜 and ℬ are Banach algebras with common identity and ℬ ⊆𝒜. Show σℐ(A) ⊆σℬ(A) and ∂σ𝒟(A)

Suppose 𝒜 and ℬ are Banach algebras with common identity and...

Suppose $\mathscr{A}$ and $\mathscr{B}$ are Banach algebras with common identity and $\mathscr{B} \subseteq \mathscr{A}$. Show $\sigma_{\mathscr{I}}(A) \subseteq \sigma_{\mathscr{B}}(A)$ and $\partial \sigma_{\mathscr{D}}(A) \subseteq \partial \sigma_{\mathscr{I}}(A)$, for any $A \in \mathscr{B}$. Hint for the second part: Since the first part implies that the interior of $\sigma_{\sigma f}(A)$ is contained in the interior of $\sigma_{\mathscr{B}}(A)$ for any $A$ in $\mathscr{B}$, argue first that it suffices to show that if $\lambda \in \partial \sigma_{\mathscr{D}}(A)$, then $\lambda \in \sigma_{\sigma d}(A)$.

Key Concepts

Banach Algebra

A Banach algebra is a complete normed algebra over the real or complex numbers with a sub-multiplicative norm that also respects the algebra multiplication. This context often considers algebras with an identity element, which is crucial for defining concepts such as the spectrum of an element.

Spectrum

In the context of Banach algebras, the spectrum of an element is the set of scalars for which the element, when subtracted by that scalar multiplied by the identity, is not invertible. This fundamental concept provides deep insights into the structure and behavior of operators or elements within the algebra.

Subalgebra Inclusion

When one Banach algebra is a subalgebra of another and shares the same identity, many properties, especially those related to invertibility, can be compared. Understanding how spectra relate under such inclusions is key to studying the variational behavior of elements between different algebras.

Spectral Inclusion

The idea of spectral inclusion refers to the principle that the spectrum of an element considered within a larger algebra may be larger or equal to its spectrum in a subalgebra. This inclusion is used to relate the spectral properties between different algebras and is central to the problem of showing that one spectrum is a subset of another.

Spectral Boundary and Interior

The spectral boundary refers to the boundary points of the spectrum, while the interior refers to its open subset. Analyzing the relationships between these, such as proving that the boundary in a subalgebra’s spectrum lies within the spectrum of a larger algebra, is essential for studying how the spectrum behaves under various algebraic manipulations and embeddings.

Transcript

00:01 Suppose g alpha, the sequence g alpha, where alpha is contained in a, a is an index set.

00:16 Suppose this is a sequence, this is a sequence or a family of sigma algebra.

00:26 We want to show that h, which is defined as the intersection, i mean, arbitrary intersection of alpha, g alpha is sigma algebra.

00:48 Okay, first we want to show omega is contained in h.

00:58 Okay, and notice for any r in a, as g alpha is a sigma algebra, we have omega must be an element in g alpha because this is the property for a sigma algebra.

01:23 Okay, now we know omega is, the capital omega is an element for any alpha.

01:32 That means if we intersecting all of those g alpha together, and omega is still contained in the intersection of them, that means omega is an element in alpha in a, g alpha.

01:53 Okay, so omega is contained in h.

01:58 Second, let's say for any set b, which is contained in h, we want to show the complement of b is also contained in h.

02:10 Okay, by the definition, h is equal to the intersection.

02:15 So we know the set b is actually contained in this intersection.

02:22 And this is equivalent to say for any alpha, b is contained in g alpha.

02:32 And again, as g alpha is a sigma algebra, we know this tells us for this alpha, the complement of b is also contained in g alpha.

02:50 Notice our alpha is arbitrary.

02:53 That means for any alpha, we always have the complement of b is contained in g alpha...