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

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.