Subvarieties of the tetrablock and von Neumann's inequality

We show an interplay between the complex geometry of the tetrablock double-struck E and the commuting triples of operators having double-struck E as a spectral set. We prove that double-struck E as a three-dimensional domain does not have any two-dimensional distinguished variety, and that every distinguished variety in the tetrablock is one dimensional and can be represented as (∗) Ω = {(x₁, x₂, x₃) ∈ double-struck E : (x₁, x₂) ∈ σ_T (A₁∗ + x₃A₂, A₂∗ + x₃A₁)}, where A₁, A₂ are commuting matrices of the same order satisfying the equality [A₁∗, A₁] = [A₂∗, A₂] and a norm condition. The converse also holds: that is, a set of the form (∗) is always a distinguished variety in double-struck E. We show that for a triple of commuting operators Y = (T₁, T₂, T₃) having double-struck E as a spectral set, there is a one-dimensional subvariety Ω_Y of double-struck E depending on Y such that von-Neumann's inequality holds; that is, f (T₁, T₂, T₃) ≤ sup _{(x1,x2,x3)∈ΩY} |f (x₁, x₂, x₃)| for any holomorphic polynomial f in three variables, provided that T₃ⁿ → 0 strongly as n → ∞. The variety Ω_Y has been shown to have representation like (∗), where A₁, A₂ are the unique solutions of the operator equations T₁-T₂∗T₃ = (I-T₃∗T₃)^1/2X₁(I-T₃∗T₃)^1/2 and T₂-T₁∗T₃ = (I-T₃∗T₃)^1/2X₂(I-T₃∗T₃)^1/2. We also show that under certain conditions, Ω_Y is a distinguished variety in double-struck E. We produce an explicit dilation and a concrete functional model for such a triple (T₁, T₂, T₃) in which the unique operators A₁, A₂ play the main role. Also, we describe a connection of this theory with the distinguished varieties in the bidisc and in the symmetrized bidisc.