Let $X$ be Hilbert spaces $\mathbb{C}^d$, and $L(X)$ be the sets of linear operators of $X$. We are given a matrix subspace $S\subset L(X)$. Via the following procedure, one can generate the smallest matrix algebra containing $S$, where $T\subset L(X)$ is called an algebra if it is closed under matrix addition and matrix multiplication.
For subspaces $A,B\subset L(X)$, $span\{A,B\}$ denotes the subspace spanned by the linear combinations of elements from $A$ and $B$.
For subspaces $A,B\subset L(X)$, $$A\cdot B=span\{ab:a\in A,b\in B\}.$$
Let $$T_0=S, T_1=span\{T_0,S\cdot T_0\},\cdots, T_n=\{T_{n-1},S\cdot T_{n-1}\}.$$
One can observe that $T_{d^2}=T_t$ for all $t>d^2$ by considering the dimension of the subspace. The question is that is $d^2$ necessary?
Moreover, is $T_d$ always an algebra? Notice that it is the case that $S$ is one dimensional.
