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= ξ ∈ G : ξ ^2 = 1 the space of reflectionsand P = Q ∩ U. For any positive a ∈ G consider the a-unitary group U_a ={ g ∈ G : a^-1 g^* a = g ^-1}, i.e. the elements which are unitary with respect to the scalar product ⟨ ξ,n ⟩ a = ⟨ a ξ,n ⟩ for ξ, n ∈ H. If π denotes the map that assigns to each invertible element its unitary part in the polar decomposition, it is shown that the restriction π |_Ua : Ua →U is a diffeomorphism, that π ( ua ∩ Q) = P and that π (Ua ∩ G^s ) = Ua ∩ G^s = { u ∈ G : u = u^* = u^-1 and au = ua }.