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 C3C3 defined asE={(x1,x2,x3)∈C3:1−zx1−wx2+zwx3≠0whenever |z|≤1,|w|≤1}. A commuting triple of operators (T1,T2,T3)(T1,T2,T3) for which the closed tetrablock E‾ is a spectral set, is called an EE-contraction. For an EE-contraction (T1,T2,T3)(T1,T2,T3), the two operator equationsT1−T2⁎T3=DT3X1DT3 and T2−T1⁎T3=DT3X2DT3,DT3=(I−T3⁎T3)12, have unique solutions A1A1, A2A2 on DT3=Ran‾DT3 and they are called the fundamental operators of (T1,T2,T3)(T1,T2,T3). For a particular class of EE-contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A1A1, A2A2 satisfyequation(0.1)A1A2=A2A1 and A1⁎A1−A1A1⁎=A2⁎A2−A2A2⁎. Then we construct an EE-contraction from that particular class which fails to satisfy (0.1). We produce a concrete functional model for pure EE-isometries, a class of EE-contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model.