On https://en.wikipedia.org/wiki/Calkin_algebra, "The Gelfand-Naimark-Segal construction implies that the Calkin algebra is isomorphic to an algebra of operators on a nonseparable Hilbert space."
Let $H$ be an infinite dimensional, complex, separable Hilbert space. By applying the Gelfand–Naimark-Segal Construction we can obtain an injective *-homomorphism $\phi: B(H)/K(H) \rightarrow B(H')$ for some Hilbert space $H'$, where $B(H)/K(H)$ is the Calkin algebra. I want to show that $H'$ here is not separable in this case.
Let's try proving it by contradiction.
Suppose there exists a countable dense subset of $H'$. And let $\{ e_n\}_{n\in \mathbb{Q}}$ be an orthonormal basis for $H$.
I know that there exists uncountable many infinite subsets $\{E_{\alpha}\}_{\alpha \in I}$ of $\mathbb{Q}$ such that $E_{\alpha } \cap E_{\beta}$ is finite for $\alpha \ne \beta$.
I was suggested to consider the orthogonal projections $P_{\alpha}$ of $H$ onto span $\{e_n: n \in E_{\alpha}\}$… But I'm not sure how to proceed from here.
Any references or suggestions will be appreciated!
Thank you!
Best Answer
You are basically done. Take $$ P_\alpha=\sum_{n\in E_\alpha}e_n. $$ If $\pi:B(H)\to B(H)/K(H)$ is the quotient map, you have $\pi(P_\alpha P_\beta)=0$ when $\alpha\ne \beta$ since $P_\alpha P_\beta$ is finite-rank.
So $\{\pi(P_\alpha)\}_\alpha$ is an uncountable family of pairwise orthogonal projections in $B(H)/K(H)$.