Polar decomposition under perturbations of the scalar product

Abstract

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, G^s the set of invertible selfadjoint elements of A, Q={e in G : e^2 = 1} the space of reflections and P = Q\cap U. For any positive a in G consider the a-unitary group U_a={g in G : a^{-1} g^* a = g^{-1}}, i.e. the elements which are unitary with respect to the scalar product <\xi,\eta>_a = <a \xi,\eta> for \xi, \eta in H. If \pi denotes the map that assigns to each invertible element its unitary part in the polar decomposition, we show that the restriction \pi|_{U_a}: U_a \to U is a diffeomorphism, that \pi(U_a \cap Q) = P and that \pi(U_a\cap G^s) = U_a\cap G^s = {u in G: u=u^*=u^{-1} and au = ua}.