[math] \newcommand{\mathds}{\mathbb}[/math]

Develop a theory of projections, isometries and symmetries inside [math]B(H)[/math], notably by examining the validity of the formula

[[math]] \lim_{n\to\infty}(PQ)^n=P\wedge Q [[/math]]

when talking about projections, and also by taking into account the fact that

[[math]] UU^*=1\iff U^*U=1 [[/math]]

does not necessarily hold in infinite dimensions, when talking about isometries.