Consider ℓ^∞ as a commutative unital C^*-algebra with product {xn} ·{yn}= {xn yn} and involution

Consider ℓ^∞ as a commutative unital C^*-algebra with...

Consider $\ell^{\infty}$ as a commutative unital $C^{*}$-algebra with product $\left\{x_{n}\right\} \cdot\left\{y_{n}\right\}=$ $\left\{x_{n} y_{n}\right\}$ and involution $\left\{x_{n}\right\}^{*}=\left\{\overline{x_{n}}\right\}$. The cluster set at infinity of $x=\left\{x_{n}\right\} \in \ell^{\infty}$ is the set $$ \begin{aligned} C l_{\infty}(x) \equiv &\{\lambda \in \mathbb{C}: \text { for every } \varepsilon>0 \text { and } N \in \mathbb{N}\\ &\text { there exists } \left.n \geq N \text { with }\left|x_{n}-\lambda\right|<\varepsilon\right\} \\ =&\left\{\lambda \in \mathbb{C}: \text { there exists a subsequence }\left\{x_{n_{k}}\right\}\right. \text { of }\\ &\left.\left\{x_{n}\right\} \text { with } x_{n_{k}} \rightarrow \lambda \text { as } k \rightarrow \infty\right\} . \end{aligned} $$ Show that for any $x=\left\{x_{n}\right\} \in \ell^{\infty}$, $\sigma(x)=\left\{x_{n}: n \in \mathbb{N}\right\} \cup C l_{\infty}(x)=\overline{\left\{x_{n}: n \in \mathbb{N}\right\}}$

Consider $\ell^{\infty}$ as a commutative unital $C^{*}$-algebra with product $\left\{x_{n}\right\} \cdot\left\{y_{n}\right\}=$ $\left\{x_{n} y_{n}\right\}$ and involution $\left\{x_{n}\right\}^{*}=\left\{\overline{x_{n}}\right\}$. The cluster set at infinity of $x=\left\{x_{n}\right\} \in \ell^{\infty}$ is the set
$$
\begin{aligned}
C l_{\infty}(x) \equiv &\{\lambda \in \mathbb{C}: \text { for every } \varepsilon>0 \text { and } N \in \mathbb{N}\\
&\text { there exists } \left.n \geq N \text { with }\left|x_{n}-\lambda\right|<\varepsilon\right\} \\
=&\left\{\lambda \in \mathbb{C}: \text { there exists a subsequence }\left\{x_{n_{k}}\right\}\right. \text { of }\\
&\left.\left\{x_{n}\right\} \text { with } x_{n_{k}} \rightarrow \lambda \text { as } k \rightarrow \infty\right\} .
\end{aligned}
$$
Show that for any $x=\left\{x_{n}\right\} \in \ell^{\infty}$,
$\sigma(x)=\left\{x_{n}: n \in \mathbb{N}\right\} \cup C l_{\infty}(x)=\overline{\left\{x_{n}: n \in \mathbb{N}\right\}}$

Key Concepts

C*-Algebra

A C*-algebra is a Banach algebra equipped with an involution satisfying the C*-identity. Such algebras provide a framework for the analysis of operators, connecting algebraic and topological properties. In the commutative unital case, the Gelfand representation theorem shows that every C*-algebra can be represented as an algebra of continuous functions on its spectrum.

Spectrum of an Element

The spectrum of an element in a C*-algebra consists of all complex numbers ? for which the element, when shifted by ? times the identity, fails to be invertible. In commutative C*-algebras, the spectrum of an element coincides with the set of its values under the Gelfand transform, providing crucial insight into its functional behavior.

Cluster Points and Cluster Sets

Cluster points are accumulation points of a sequence, meaning any value that is approached by some subsequence. The cluster set at infinity specifically gathers all limit points of subsequences of a bounded sequence, reflecting the asymptotic behavior of the sequence as it extends indefinitely.

Closure of a Set

In topology, the closure of a set is the smallest closed set containing it, including all its limit points. This concept is important in analysis since many properties of sequences and functions are best understood by considering the closure of sets, which incorporates not only the original elements but also all points that can be approached arbitrarily closely by them.

Transcript

00:01 In this problem, we have an absolutely converging series a -7, n, and we have another series whose nth term is any rearrangement of the original series a -7.

00:20 Then we want to show that the series of rearrangement converges, and it has the same sum of the original series a -7.

00:34 So first let's note that if the series a's of n converges absolutely, then it converges.

00:59 Then the series converges and we can say a real number l is equal to the series, the sum of the entire series.

01:22 Now let's say that s of k is any partial sum of the series, then using the definition of convergence of a series, the limit when k goes to infinity of s sub k is equal to l.

02:03 That is, if we take any positive number epsilon, then there exists a natural number, let's say n, such that we are writing it here, such that if i is greater than or equal to n, then the absolute value of the partial sum s of i minus l is less than epsilon half, which is a positive quantity.

03:00 So here we are using definition of convergence of the sequence of partial sum to the value l, which is the sum of the entire series.

03:17 So now we can use the fact that the series of absolute values converges, then we can say that there exists number, let's say, m, natural number m such that if i is squared than are equal to m, then the absolute value of the partial sum of the series of absolute values, a x of n, minus the entire series is less than epsilon half.

04:38 And that's again the definition that the partial sums of this convergent series of the absolute values, 8 of n converges to the series.

04:57 And now we can rewrite this this way.

05:02 The absolute value of the series from n equals i plus 1 to infinity, of the absolute value of h of n is less than epsilon half.

05:17 That's because we can, we are taking out from the entire series the sum of the first i terms.

05:25 And that is equal then to the series, starting at i i plus 1 and here since the series is of absolute values is always non -negative so we can take out we can get rid of the external absolute value so this is equal to the series from n equals i plus 1 to infinity of the absolute value of a x of n and that's less than epsilon half and that's true for any index i greater than or equal to m for some natural number m so we have these two facts this one here and this one here we are going to combine these two to prove the first first part of the problem so now we can take in particular you take i equals big m we have that the series from big m plus one to infinity of the absolute value of 8 of n is less than xlome half.

06:52 That's because we are taking here just i equal big m.

06:58 So we have this expression here for i equals big m.

07:03 It must be true.

07:10 Now we can take n1, the first index we are looking for n1 equals to m plus 1.

07:21 And so we have already that the sum from, we have already that the sum from n equals n1 to infinity of the absolute values of a sub n, less than epsilon half.

07:50 So we have already this inequality for the natural index n1.

08:02 Now we are going to choose n2.

08:04 And for that, we are going to see that we have two possibilities...

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