Extending GUE to a measure on operators? Let $\mathcal H_n$ denote the space of Hermitian $n\times n$ matrices and let $\mu_{GUE}$ denote the following measure on $\mathcal H_n$:
$$
\mu_{GUE}(dM) = \exp\left( -\frac{n}{2}\text{tr}\ M^2\right)\ dM,
$$
where $dM$ denotes the Lebesgue measure on $\mathcal H_n$. For fixed $n$,  $\mu_{GUE}$ has finite mass. The resulting normalized measure is the GUE ensemble studied in Random Matrix Theory.
Question: Can this measure be extended to the space of self-adjoint operators?
Fuzzy question: Do people study Gaussian ensembles of random self-adjoint operators?
 A: A classic problem in this connection is to derive GOE or GUE statistics for the spectrum of a random self-adjoint operator of the form $H=-\nabla^2 + V(\vec{r})$, in some bounded domain of $\mathbb{R}^3$. The measure is the Gaussian measure for $V(\vec{r})$, of zero mean and given two-point correlation function. Since this is a real operator, one would expect GOE statistics, to obtain GUE statistics one would replace $\nabla\mapsto \nabla+i\vec{B}\times\vec{r}$ for some given vector $\vec{B}$.
This problem was solved by Konstantin Efetov in 1982, as described in much detail in his book on Supersymmetry in Disorder and Chaos. The GOE or GUE statistics is found to hold only over a limited range $E_{0}$ (the so-called Thouless energy): eigenvalues that differ by more than $E_0$ become uncorrelated, reverting to Poisson statistics.
A: One way of looking at this is that, up to normalization, the GUE is standard Gaussian measure on the space of Hermitian matrices, equipped with the Hilbert-Schmidt or Frobenius inner product.  So what you're asking for is a special case of extending the notion of standard Gaussian measure to a particular infinite-dimensional space.  But you can't do that.
A: Do you mean Free Probability Theory (See e.g. Voiculescu, D. V.; Dykema, K. J.; Nica, A. Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups) ?
