Why does $w_i^*xw_j\in e\mathscr{M}e$ here?

Question

I'm reading Proposition IV.1.8 of Theory of Operator Algebras I. The statement is following:

At the last line of the proof the author said the following:

hence $(U^*xU)_{i,j}= w_i^*xw_j\in e\mathscr{M}e$...

Here $e:=w_{i_0, i_0}$ and $w_i:=w_{i, i_0}$. I cannot understand why $w_i^*xw_j\in e\mathscr{M}e$?

What I can see is if $y:=w_i^*xw_j=w_{i_0, i}xw_{j, j_0}$ so that $ey = e(w_i^*xw_j) = w_i^*xw_j=y$. Therefore the range projection $R_y$ of $y$ is a subprojection of $e$. Thus $R_y\in e\mathscr{M}e$. But why $y\in e\mathscr{M}e$?

Appreciate any explanation. Thanks.

Answer 1

Since $(w_{i,j})$ is a system of matrix units, we have $w_i=w_{i,i_0}w_{i_0,i_0}=w_i e$. Thus $w_i^\ast x w_i=ew_i^\ast xw_i e\in e\mathscr M e$.