Beijing Institute of Mathematical Sciences and Applications

Semen Artamonov , Yevgen Makedonskyi , Pavel Nikitin , Shamil Shakirov

Sohail Farhangi

Friday, May 8, 2026 1:30 PM - 3:00 PM

A3-2a-302

Zoom 242 742 6089 (BIMSA)

It is a classical consequence of the GNS construction that for any locally compact second countable group $G$, and any positive definite function $\varphi:G\rightarrow\mathbb{C}$ normalized so that $\phi(e) = 1$, there exists a Hilbert space $\mathcal{H}$, a unitary representation $U$ of $G$ on $\mathcal{H}$, and a cyclic vector $\xi$ for which $\phi(g) = \langle U_g\xi, \xi\rangle$. The Gaussian measure space construction lets us slightly refine this by taking $\mathcal{H} = L^2(X,\mu)$ for some standard probability space $(X,\mathscr{B},\mu)$, $U_g = T_g$ is the Koopman representation induced by a measure preserving action of $G$, and $\xi = f \in L^2(X,\mu)$. We will show that if $G$ is abelian, then we can further specify that $f:X\rightarrow\mathbb{S}^1$ for some measure-preserving system $(X,\mathscr{B},\mu,(T_g)_{g \in G})$. We will also show that for many non abelian groups $G$, we cannot specify that $f:X\rightarrow\mathbb{S}^1$, and for the free group $F_2$ we cannot even specify that $f \in L^\infty(X,\mu)$.