Polar decomposition under perturbations of the scalar product
نویسندگان
چکیده
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 the set of invertible selfadjoint elements of A, Q = f" 2 G : " = 1g the space of re ections and P = Q \ U . For any positive a 2 G consider the a-unitary group Ua = fg 2 G : a g a = g g, i.e., the elements which are unitary with respect to the scalar product h ; ia = ha ; i for ; 2 H. If denotes the map that assigns to each invertible element its unitary part in the polar decomposition, it is shown that the restriction jUa : Ua ! U is a di eomorphism, that (Ua \ Q) = P , and that (Ua \G) = Ua \G = fu 2 G : u = u = u 1 and au = uag:
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