Hyperfinite extensions of bounded operators on a separable Hilbert space

Abstract:

Let H be a separable Hilbert space and Ĥ the nonstandard hull of H with respect to an ${\aleph _1}$-saturated enlargement. Let S be a $^\ast$-finite dimensional subspace of $^\ast H$ such that the corresponding hyperfinite dimensional subspace Ŝ of Ĥ contains H. If T is a bounded operator on H, then an extension  of T to Ŝ where  is obtained from an internal $^\ast$-linear operator on S is called a hyperfinite extension of T. It is shown that T has a compact (selfadjoint) hyperfinite extension if and only if T is compact (selfadjoint). However T has a normal hyperfinite extension if and only if T is subnormal. The spectrum of a hyperfinite extension  equals the point spectrum of Â, and if T is quasitriangular, A can be chosen so that the spectrum of  equals the spectrum of T. A simple proof of the spectral theorem for bounded selfadjoint operators is given using a hyperfinite extension.