the sum of projections is 1 in $B(H)$

Question

Suppose $H$ is an infinite dimensional Hilbert space,can we find $n $ isometries $s_1,s_2,\cdots,s_n \in B(H)$ such that $1=s_1s_1^*+s_2s_2^*+\cdots s_ns_n^*$ ,where $1$ is the identity map from $H$ to $H$.

Answer 1

Indeed we can. Moreover these isometries will generate the well known Cuntz algebra (https://en.wikipedia.org/wiki/Cuntz_algebra).

You can take an orthonormal basis $(e_k)_{k=1}^{\infty}$ for $\mathcal{H}$ (assuming $\mathcal{H}$ is separable here). Now partition $\mathbb{N} = N_1 \sqcup \cdots \sqcup N_n$ where $N_j = \{j + kn \mid k \geq 0\}$. Letting $f_j: \mathbb{N} \to N_j$ be bijections, we can let $S_je_k = e_{f_j(k)}$, extend by linearity and continuity. Then you can check that $(S_j)_{j=1}^n$ are $n$ isometries satisfying your relation (called the Cuntz relation).