ELA, Volume 7, pp. 21-29, February 2000, abstract.

Polar Decomposition under Perturbations of the Scalar Product

Gustavo Corach, Alejandra Maestripieri, and Demetrio Stojanoff


Let A be a unital C*-algebra with involution * represented
in a Hilbert space H, G the group of invertible elements
of A, U the unitary group of A, Gs the set of invertible
selfadjoint elements of A, Q the space of reflections and P the
space of unitary (or, equivalently, selfadjoint) reflections.
For any positive element a of G, consider the a-unitary group
Ua of elements of A which are unitary with respect to the
scalar product induced by a, i.e. the product
(x, y) = <ax, y>, for x, y in H, where < , > is the usual
scalar product of H.
 
If p denotes the map  that assigns to each invertible element
its unitary part in the polar decomposition, it is  shown that
the restriction of p to Ua: p: Ua --> U is a diffeomorphism,
that p(Ua intersection Q) = P and that 
p(Ua intersection Gs) = Ua intersection Gs.