## Polar decomposition under perturbations of the scalar product

### Summary

Summary: Let A be a unital C $\Lambda$-algebra with involution $\Lambda$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 = f" 2 G : " 2 = 1g the space of reflections and P = Q " U. For any positive a 2 G consider the a-unitary group U a = fg 2 G : a $\Gamma 1$ g $\Lambda a = g \Gamma 1$ g, i.e., the elements which are unitary with respect to the scalar product h; ji a = ha; ji for ; j 2 H. If ss denotes the map that assigns to each invertible element its unitary part in the polar decomposition, it is shown that the restriction ssj U a : U a ! U is a diffeomorphism, that $ss(U a " Q) = P$ , and that $ss(U a " G s ) = U$ a " G s = fu 2 G : u = u $\Lambda = u \Gamma 1$ and au = uag:

47A30, 47B15

### Keywords/Phrases

polar decomposition, C $\Lambda$-algebras, positive operators, projections