View source for GoI for MELL: partial isometries

= The proof space =

Given a partial permutation <math>\varphi</math> one defines a partial isometry <math>u_\varphi</math> by:
: <math>u_\varphi(e_n) = 
   \begin{cases}
     e_{\varphi(n)} & \text{ if }n\in D_\varphi,\\
     0              & \text{ otherwise.}
   \end{cases}
  </math>
In other terms if <math>x=(x_n)_{n\in\mathbb{N}}</math> is a sequence in <math>\ell^2</math> then <math>u_\varphi(x)</math> is the sequence <math>(y_n)_{n\in\mathbb{N}}</math> defined by:
: <math>y_n = x_{\varphi^{-1}(n)}</math> if <math>n\in C_\varphi</math>, <math>0</math> otherwise.

We will (not so abusively) write <math>e_{\varphi(n)} = 0</math> when <math>\varphi(n)</math> is undefined so that the definition of <math>u_\varphi</math> reads:
: <math>u_\varphi(e_n) = e_{\varphi(n)}</math>.

The domain of <math>u_\varphi</math> is the subspace spanned by the family <math>(e_n)_{n\in D_\varphi}</math> and the codomain of <math>u_\varphi</math> is the subspace spanned by <math>(e_n)_{n\in C_\varphi}</math>. In particular if <math>\varphi</math> is <math>1_D</math> then <math>u_\varphi</math> is the projector on the subspace spanned by <math>(e_n)_{n\in D}</math>.

{{Definition|
We call ''<math>p</math>-isometry'' a partial isometry of the form <math>u_\varphi</math> where <math>\varphi</math> is a partial permutation on <math>\mathbb{N}</math>. The ''proof space'' <math>\mathcal{P}</math> is the set of all <math>p</math>-isometries.
}}

{{Proposition|
Let <math>\varphi</math> and <math>\psi</math> be two partial permutations. We have:
: <math>u_\varphi u_\psi = u_{\varphi\circ\psi}</math>.

The adjoint of <math>u_\varphi</math> is:
: <math>u_\varphi^* = u_{\varphi^{-1}}</math>.

In particular the initial projector of <math>u_{\varphi}</math> is given by:
: <math>u_\varphi u^*_\varphi = u_{1_{D_\varphi}}</math>.

and the final projector of <math>u_\varphi</math> is:
: <math>u^*_\varphi u_\varphi = u_{1_{C_\varphi}}</math>.

If <math>p</math> is a projector in <math>\mathcal{P}</math> then there is a partial identity <math>1_D</math> such that <math>p= u_{1_D}</math>.

Projectors commute, in particular we have:
: <math>u_\varphi u_\varphi^*u_\psi u_\psi^* = u_\psi u_\psi^*u_\varphi u_\varphi^*</math>.
}}

Note that this entails all the other commutations of projectors: <math>u^*_\varphi u_\varphi u_\psi u^*_\psi = u_\psi u^*_\psi u^*_\varphi u_\varphi</math> and <math>u^*_\varphi u_\varphi u^*_\psi u\psi = u^*_\psi u_\psi u^*_\varphi u_\varphi</math>.

In particular note that <math>0</math> is a <math>p</math>-isometry. The set <math>\mathcal{P}</math> is a submonoid of <math>\mathcal{B}(H)</math> but it is not a subalgebra.<ref><math>\mathcal{P}</math> is the normalizing groupoid of the maximal commutative subalgebra of <math>\mathcal{B}(H)</math> consisiting of all operators ''diagonalizable'' in the canonical basis.</ref>In general given <math>u,v\in\mathcal{P}</math> we don't necessarily have <math>u+v\in\mathcal{P}</math>. However we have:

{{Proposition|
Let <math>u, v\in\mathcal{P}</math>. Then <math>u+v\in\mathcal{P}</math> iff <math>u</math> and <math>v</math> have disjoint domains and disjoint codomains, that is:
: <math>u+v\in\mathcal{P}</math> iff <math>uu^*vv^* = u^*uv^*v = 0</math>.
}}

{{Proof|
Suppose for contradiction that <math>e_n</math> is in the domains of <math>u</math> and <math>v</math>. There are integers <math>p</math> and <math>q</math> such that <math>u(e_n) = e_p</math> and <math>v(e_n) = e_q</math> thus <math>(u+v)(e_n) = e_p + e_q</math> which is not a basis vector; therefore <math>u+v</math> is not a <math>p</math>-permutation.
}}

As a corollary note that if <math>u+v=0</math> then <math>u=v=0</math>.