2018/05/27 by Chi-Kwong Li, Vern I. Paulsen, Li, Chi-Kwong +3 · 2 citations
Mathematics · #47A12 #47A13 #47A20 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1805.10600
openalex publication_date 2018/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A = (A1, …, Am) be an m-tuple of elements of a unital C*-algebra \cal A and let Mq denote the set of q × q complex matrices. The joint q-matricial range Wq(A) is the set of (B1, …, Bm) ∈ Mqm such that Bj = Φ(Aj) for some unital completely positive linear map Φ: \cal A → Mq. When \cal A= B(H), where B(H) is the algebra of bounded linear operators on the Hilbert space H, the \bf joint spatial q-matricial range Wqs(A) of A is the set of (B1, …, Bm) ∈ Mqm for which there is a q-dimensional V of H such that Bj is a compression of Aj to V for j=1,…, m. Suppose K(H) is the set of compact operators in B(H). The joint essential spatial q-matricial range is defined as Wessq(A) = ∩ \ \bf cl(Wsq(A1+K1, …, Am+Km)): K1, …, Km ∈ K(H) \, where \bf cl denotes the closure. Let π be the canonical surjection from B(H) to the Calkin algebra B(H)/K(H). We prove that Wessq(A) =Wq(π(A) , where π(A) = (π(A1), …, π(Am)). Furthermore, for any positive integer N, we prove that there are self-adjoint compact operators K1, …, Km such that \bf cl(Wqs(A1+K1, …, Am+Km)) = Wqess(A) \hbox for all q ∈ \1, …, N\. These results generalize those of Narcowich-Ward and Smith-Ward, obtained in the m=1 case, and also generalize a result of Müller obtained in case m ≥ 1 and q=1. Furthermore, if Wess1(\bf A) is a simplex in \mathbb Rm, then we prove that there are self-adjoint K1, …, Km ∈ K(H) such that \bf cl(Wqs(A1+K1, …, Am+Km)) = Wqess(A) for all positive integers q.