Let $H$ be a Hilbert space, $S(H)$ be the inverse semigroup (pseudogroup) of linear maps between (closed) subspaces of $H$ preserving the dot product (the operation is composition of partial maps). Now consider an arbitrary inverse semigroup $A$ and all possible homomorphisms $A\to S(H)$. An inverse semigroup, by definition, is  a semigroup with unary operation $^{-1}$ such that $(a^{-1})^{-1}=a, aa^{-1}a=a, aa^{-1} bb^{-1}=bb^{-1} aa^{-1}$ (by Wagner's theorem, similar to the Cayley theorem for groups) these are precisely the semigroups of partial bijections between subsets of a set under composition). 

1. Is it possible to define property (T) for inverse semigroups 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)? 

Note that amenability of inverse semigroups (pseudogroups) has been considered, for example, <a href="http://www.google.com/url?sa=t&source=web&cd=4&ved=0CCAQFjAD&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.54.4758%26rep%3Drep1%26type%3Dpdf&rct=j&q=property%20(T)%2C%20pseudogroups&ei=htSgTf2nKNCztwfPn5n7Ag&usg=AFQjCNEFr3TB1ZY1aft6xxUu8GWdvjAUxw">here. </a> 

<b>Update</b> Since there is a confusion about what partial maps are, here is a formal definition. A partial map $X\to X$ is a map from a subset $Y$ of $X$, called domain, to another subset $Z\subseteq X$, called range. If $X$ is a metric space (say, a Hilbert space), a partial isometry is an isometry between subspaces $Y$ and $Z$. The composition of a partial map $f: X\to X$ and another partial map $g: X\to X$ is a partial map $fg$ defined of all $x$ such that $x$ is in the domain of $g$ and $g(x)$ is in the domain of $f$, $fg(x)=f(g(x))$. If $f:Y\to Z$ is a (partial) bijection $X\to X$, then the map  $f^{-1}f$ is the identity map on the domain of $f$, and it is an idempotent (obviously). If $f$ is an idempotent partial bijection with damain (=range) $Y$, and $g$ is an idempotent with domain (=range) $Z$, then $fg=gf$ is the idempotent with domain (=range) $Y\cap Z$, so idempotents commute. It is obvious that any set of partial bijections closed under products and taking $^{-1}$ is an inverse semigroup. The question is about representations of inverse semigroups into the inverse semigroup of partial (!) bijective unitary (=preserving the dot product) operators of a Hilbert space.