Calkin algebra and $\beta\omega\setminus\omega$.

Question

Recall that the Calkin algebra is defined as the quotient $\mathcal{B}(\ell^2)/\mathcal{K}(\ell^2)$ where $\mathcal{B}(\ell^2)$ is the algebra of bounded operators and $\mathcal{K}(\ell^2)$ the ideal of compact ones.

On the other hand $\omega$ is the set of natural numbers and $\beta\omega$ its $\check{C}$ech-$S$tone compactification.

Reading about Calkin algebra I found the following expression:

$\textbf{The Calkin algebra is the non-commutative analog to }\beta\omega\setminus\omega.$

Somebody knows in what sense is such an analogy given or where can I find some bibliography about it?

Thank you!

Can you provide a reference on what the Calkin algebra is? The value of your question would increase manyfold if it were useful for others, too. — Mariano Suárez-Álvarez, Apr 28 '17 at 20:10

Tomasz Kania · Accepted Answer · 2017-04-29T11:55:48.900

If you accept the analogies

$$c_0 \leftrightarrow \mathcal{K}(\ell_2),$$ $$\ell_\infty \leftrightarrow \mathcal{B}(\ell_2),$$

then you also have $$C(\beta\mathbb N \setminus\mathbb N) \leftrightarrow \mathcal{B}(\ell_2)/\mathcal{K}(\ell_2),$$

as $C(\beta \mathbb N\setminus \mathbb N)\cong \ell_\infty / c_0$ and $C(\beta \mathbb N\setminus \mathbb N)$ is encoded by its spectrum $\beta \mathbb N\setminus \mathbb N$ (by the Gelfand-Kolmogorov theorem, for example).

The algebra $\ell_\infty$ is the canonical diagonal masa in $\mathcal{B}(\ell_2)$ and so $\ell_\infty / c_0$ is the canonical diagonal masa in the Calkin algebra. Thus, you have one more analogy.

+1. I would add to your answer that you can see $c_0$ as $C_0(\mathbb N)$, and $\ell_\infty$ as $C(\beta\mathbb N)$. — Martin Argerami, Apr 28 '17 at 23:23

Calkin algebra and $\beta\omega\setminus\omega$.

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