Pure-state homogeneity of the completed CAR algebra

Let be the completed CAR algebra, the norm completion of the matrix stages under , with the fixed coordinate reindexing. A state means a positive continuous complex-linear functional taking value at the identity; a pure state is an extreme point of this convex state space. Given any pure states on , there is a complex star automorphism such that for every . For every finite set and every , a unitary can be chosen with for all .

The algebra is the fixed CAR completion, and the functionals are pure states. There is no additional intertwining-sequence hypothesis and no general Kishimoto–Ozawa–Sakai assumption.

The approximately inner automorphism transports to in the direction . It is surjective because the construction controls actual inverse automorphisms as well as forward maps.

The pure-state sequence theorem provides inner automorphisms with forward and inverse pointwise Cauchy behavior and the transported-state limit. The conditional homogeneity theorem applies its two-sided limit construction to this supplier. These two results are exactly the dependencies composed in the selected proof.

This theorem asserts homogeneity for CAR. It does not prove the separately defined statement for all simple separable C*-algebras. Approximate innerness does not assert that the two GNS representations are already unitarily equivalent. The adjacent asymptotically inner result carries extra path data and is not the direct proof used by this declaration.