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

Abstract

Abstract.Let Ω ⊂ C n be a bounded convex or strictly pseudoconvex open subset. Given a separable Hilbert space K and a weak * closed subspace T ⊂ B(K), we show that the space H ∞ (Ω, T ) of all bounded holomorphic T -valued functions on Ω possesses the tensor product representation H ∞ (Ω, T ) = H ∞ (Ω)⊗T with respect to the normal spatial tensor product. As a consequence we deduce that H ∞ (Ω) has property S σ . This implies that, if S ∈ B(H) n is a subnormal tuple of class A on a strictly pseudoconvex or bounded symmetric domain and T ∈ B(K) m is a commuting tuple satisfying AlgLat(T ) = A T (where A T denotes the unital dual operator algebra generated by T ), then the tensor product tuple (S ⊗ 1, 1 ⊗ T ) is reflexive.