On Algebras of Functions over Infinite Dimensions

Abstract

We introduce a family of reproducing kernel Hilbert spaces $\mathcal A_\Lambda$ of holomorphic functions defined on an infinite--dimensional domain in a separable Hilbert space, $\mathbb{H}$. The reproducing kernel of $\mathcal A_\Lambda$ is constructed using the covariance operator associated with a Gaussian measure on $\mathbb{H}$, along with a holomorphic function $\Lambda$ on the unit disk. Under certain conditions on the kernel, $\mathcal A_\Lambda$ is closed under pointwise multiplication, giving it the structure of a reproducing kernel Hilbert algebra (RKHA). We also study twisted canonical commutation relations on these RKHAs, where the creation and annihilation operators are both bounded.