I am studying the Cuntz algebra $\mathcal{O}_n$, $(n \ge 2)$ with generators $S_1, S_2, \ldots, S_n$ and in my class notes there is a statement about the projections $S_1S_1^*, S_2S_2^*, \ldots, S_nS_n^*$ which seems to be easy but I haven't been able to prove:
- All the projecions $S_1S_1^*, S_2S_2^*, \ldots, S_nS_n^*$ are infinite, where a projection $p$ in a C*-algebra $\mathcal{A}$ is infinite if it is equivalent to a proper subprojection $q \in \mathcal{A}$ in the sence that there existis $v \in A$ such that $p=v^*v$ and $q=vv^*$.
I proved that the identity $1$ is an infinite projection, but the projections in the statement seems to be trickier.
If someone can help me, I appreciate.
They are equivalent to the identity, so they are also infinite.
Explicitly, note that $S_jS_jS_j^*S_j^*$ is a subprojection of $S_jS_j^*$.
It is proper, because if $S_jS_jS_j^*S_j^*=S_jS_j^*$, multiplying by $S_j^*$ on the left and $S_j$ on the right we get $S_jS_j^*=I$, a contradiction.
And it is equivalent to $S_jS_j^*$, because if $V=S_jS_jS_j^*$, then $V^*V=S_jS_j^*$ and $VV^*=S_jS_jS_j^*S_j^*$.