Let $H$ be a Hilbert space, $S(H)$ be the inverse semigroup (groupoid) of linear maps between subspaces of $H$ preserving the dot product (the operation is composition). Now consider an arbitrary inverse semigroup $A$ and all possible representations $A\to S(H)$.

1. Is it possible to define property (T) for inverse semigroups (groupoids) using these representations?

2. Let $I_n(\mathbb Z)$ be the set of restrictions of all operators from $SL_n(\mathbb Z)$ onto all subspaces of $\mathbb{R}^n$, $n\ge 3$. It is an inverse semigroup. Does it have property (T)?