2025/09/22 by Pankaj Dey, Dey, Pankaj, Atanu Dhang +3
Mathematics · #46L07 #46L52 #47A12 #47A20 #47L25 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematics and Applications #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2509.18027
openalex publication_date 2025/09/22 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
Let Jn be the Jordan block of size n with all eigen values 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). Later, Choi and Li generalize this result of Arveson and prove that every unital positive map on the operator system generated by any 2× 2 matrix or any 3×3 matrix with a reducing subspace is completely positive. 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. We begin by establishing a structure theorem for a dilation of an operator B satisfying BB^*+B^*B=I and then investigate whether every B∈\mathbbMn satisfying BB^*+B^*B≤ In admits a dilation \widetildeB for which \widetildeB\widetildeB^*+\widetildeB^*\widetildeB=I. This study plays the central role to the development of this paper. We use this to prove that every unital contractive map on the operator system generated by J3 is 2-positive and obtain some partial results towards characterizing the matricial range of J3. 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 \Re(λ)≥ 0. We prove that every unital contractive map on the operator system generated by T=diag(1,-1,i,-i) is completely positive.