Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume that the shifts act ergodically). There are many examples of those, the best known are probably the Gaussians with their Cameron-Martin space of shifts. Others include even "Lebesgue" examples, i.e. $\sigma$-finite measures that are actually invariant under large subspaces of shifts.
For any such measure type there is a canonical unitary representation of $S$ in the space of square roots of measures absolutely continuous w.r.t. $M$. By the spectral theorem, there is a "Fourier transform" between $L^2(M)$ (or better, square roots of measures) and a direct integral over the spectral measure type $\hat M$, over which these shifts turn into measurable linear functionals.
This idea seems pretty much standard, so I assume it should have been worked out in the literature, but I couldn't find that. Specifically, I'm interested in the following questions:
Under which conditions is the duality between $M$ and $\hat M$ reflexive (and the spectral multiplicity is $1$)? Is there a corresponding duality between convolution and multiplication? The latter isn't even well-defined for square roots of measures, while the former is, at least at the level of measure types.
Are there any reasonable connections between this "$L^2$ theory" and the "$L^1$ theory", i.e. Fourier transform of measures?
Do you know of any references on that?
