The failure of rational dilation on the tetrablock

We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C3 defined asE={(x1,x2,x3)∈C3:1-zx1-wx2+zwx3≠0whenever |z|≤1,|w|≤1}. A commuting triple of operators (T₁, T₂, T₃) for which the closed tetrablock E[U+203E] is a spectral set, is called an E-contraction. For an E-contraction (T₁, T₂, T₃), the two operator equationsT1-T2*T3=DT3X1DT3 and T2-T1*T3=DT3X2DT3,DT3=(I-T3*T3)12, have unique solutions A₁, A₂ on DT3=Ran[U+203E]DT3 and they are called the fundamental operators of (T₁, T₂, T₃). For a particular class of E-contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A₁, A₂ satisfy(0.1)A1A2=A2A1 and A1*A1-A1A1*=A2*A2-A2A2*. Then we construct an E-contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure E-isometries, a class of E-contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model.