Measures on infinite dimensional Grassmann manifolds

On the support of quasi-invariant measures on infinite-dimensional Grassmann manifolds

“…Then In [2] it was shown that Prob({|a:_1|00 < oo}) = 0 (here |i_1|oo is the sup norm relative to the H+ norm for x~l acting on H*e).…”

“…Let {ejij G Z, j t¿ 0} be an orthonormal basis for H with ej G ü± when ±j > 0, so that we can identify operators from H+ to ü_ with matrices (^¿j)¿<o,i>o-Let M denote the space of all such matrices. For each s > -1 we can define a (following [2]) probability measure ps on M by either of the following two methods: (1) where dvs is the measure on S defined in §2 (here Z G M is a matrix with Ztj -0 for all but finitely many (i,j))-…”

“…For g G U(oo), because g maps cylinder sets to cylinder sets, it's easily checked that gtps = p(g, -)s dps, where p(g, Z) = detKo-1) + Kg-^Z]2 (see §2 of [2]). If g G TV, we can find gn G N such that gn -> g in N. If <p is a bounded continuous function on Gr(E), (3.5) / 4>(gn ■ z) dps(z) = / 4>(z)p(gn, z)s dps(z).…”

“…In [2] a family of finitely additive measures on Gr, {ps: s > -1}, was constructed. It was shown that (for s > 0 and ^ even integer) these measures are quasi-invariant with respect to the restricted unitary transformations in the sense of Irving Segal's algebraic integration theory (i.e., as set transformations).…”

“…In [2] we showed that x"1 is an unbounded operator with probability one. Thus this result seems reasonably sharp.…”