Matricial Ranges, Dilations, and Unital Contractive Maps

Let Jn be the Jordan block of size n with all eigenvalues zero.Arveson introduced the notion of the matricial range of an operator in his remarkable article called Subalgebras of C * -algebras II (Acta Math, 128, 1972) and established that every unital positive map on the operator system generated by J2 is completely positive.This describes the matricial range of J2 as the set of all matrices with numerical radius at most 1 2 .After fifty years of the above result of Arveson, the matricial range of Jn for n ≥ 3 has not been characterized.This article aims to investigate this long-standing open problem for n = 3.First, we prove that every unital contractive map on the operator system generated by J3 is 2-positive.Next, we show that every unital contractive map on the operator system generated by J3 is completely positive in certain cases.As a consequence, we obtain a partial description of the matricial range of J3.To establish these results, we develop a structure theorem for a dilation of an operator B satisfying BB * + B * B = I and then investigate whether every B ∈ Mn satisfying BB * + B * B ≤ In admits a dilation B for which B B * + B * B = I.We show that this has an affirmative answer for n = 2, but fails to hold in general for n ≥ 3. Next, we study unital contractive maps on operator systems generated by 4 × 4 normal matrices, and show that this is equivalent to studying a unital contractive map on the operator system generated by T = diag(λ, -1, i, -i), where ℜ(λ) ≥ 0. We prove that every unital contractive map on the operator system generated by T = diag(1, -1, i, -i) is completely positive.We conclude with some examples, remarks on the matricial range of Jn and a few applications on the constrained unitary dilations of J2. 2020 Mathematics Subject Classification.46L07,