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}.

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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.

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