properly infinite $C^*$-algebra $B(H)$

Question

There is a conclusion:If $H$ is an infinite dimensional Hibert space,then $B(H)$ is a properly infinite $C^*$-algebra.

According to the definition,we need to find mutually orthogonal projections $e,f\in B(H)$ such that $e\leq id_H,f\leq id_H$ and $e,f$ are equivalent to $id_H$.

I try to choose $e=id_H,f=UU^*$,where $U$ is an unilateral shift operator,but it doesn't satisfy the condition:$ef=fe=0$,I also attempted to define $e=e_1\otimes e_1,f=e_2\otimes e_2$,but $e$ is not equivalent to $id_H$.

My question: how to find projections $e,f$ fulfill the above conditions?

Answer 1

Let $\{e_i:i\in I\}$ be an orthonormal basis of $H$. Since $I$ is infinite, we can find two disjoint subsets $A,B\subset I$ and bijections $f_A:I\to A$, $f_B:I\to B$. Then let $e$ be the projection onto $\overline{\operatorname{span}}\{e_i:i\in A\}$, and let $f$ be the projection onto $\overline{\operatorname{span}}\{e_i:i\in B\}$. Then $e\perp f$. To show that $e\sim_{\operatorname{MvN}} 1$, define $u\in B(H)$ by linear extension of $u(e_i)=e_{f_A(i)}$. Then $u^*u=1$, and $uu^*=e$. Showing that $f\sim_{\operatorname{MvN}} 1$ is similar.