On the slice map problem for $H^\infty (\Omega )$ and the reflexivity of tensor products

Abstract:

Let $\Omega \subset \mathbb {C}^n$ be a bounded convex or strictly pseudoconvex open subset. Given a separable Hilbert space $K$ and a weak$^*$ closed subspace $\mathcal {T} \subset B(K)$, we show that the space $H^\infty (\Omega , \mathcal {T})$ of all bounded holomorphic $\mathcal {T}$-valued functions on $\Omega$ possesses the tensor product representation $H^\infty (\Omega , \mathcal {T}) = H^\infty (\Omega ){\overline {\otimes }}\mathcal {T}$ with respect to the normal spatial tensor product. As a consequence we deduce that $H^\infty (\Omega )$ has property $S_\sigma$. This implies that, if $S\in B(H)^n$ is a subnormal tuple of class $\mathbb {A}$ on a strictly pseudoconvex or bounded symmetric domain and $T \in B(K)^m$ is a commuting tuple satisfying $\text {AlgLat}(T) = \mathcal {A}_T$ (where $\mathcal {A}_T$ denotes the unital dual operator algebra generated by $T$), then the tensor product tuple $(S\otimes 1, 1 \otimes T)$ is reflexive.