The algebraic invariant r_∞(M) of a factor M, introduced in an earlier paper and called the asymptotic ratio set, is shown to be closed for any factor M. As a consequence, this set must be one of the following sets: (i) the empty set, (ii) {0}, (iii) {1}, (iv) a one parameter family of sets {0, xⁿ; n=0, ±1, …}, 0<x<1. (v) all non-negative reals, (iv) {0.1}.

View full abstract

Let E be a certain nuclear topological vector space contained in the Hilbert space (L²) on the real line and let the Gaussian probability measure be defined on the conjugate space E^* of E. We consider such a subgroup of the rotation group of (L²) that acts on E and contains the shift as a one-parameter subgroup. With a rather systematic way we define two more one-parameter subgroups which, together with the shift, constitute a subgroup isomorphic with the projective linear group PGL(2, R). It plays a role of the time change of the white noise. In this set up we formulate and prove the principle of projective invariance of the Brownian motion given by P. Lévy.

View full abstract