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Preservation of the joint essential matricial range

Preservation of the joint essential matricial range Let A=(A1,⋯,Am) be an m‐tuple of elements of a unital C∗‐algebra 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:A→Mq. When A=B(H), where B(H) is the algebra of bounded linear operators on the Hilbert space H, the joint spatial q‐matricial range Wsq(A) of A is the set of (B1,⋯,Bm)∈Mqm for which there is a q‐dimensional subspace V of H such that Bj is the compression of Aj to V for j=1,⋯,m. Suppose that K(H) is the set of compact operators in B(H). The joint essential spatial q‐matricial range is defined as Wessq(A)=∩{cl(Wsq(A1+K1,⋯,Am+Km)):K1,⋯,Km∈K(H)},where cl(T) denotes the closure of the set T. 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 clWsq(A1+K1,⋯,Am+Km)=Wessq(A)forallq∈{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(A) is a simplex in Rm, then we prove that there are self‐adjoint compact operators K1,⋯,Km such that cl(Wsq(A1+K1,⋯,Am+Km))=Wessq(A) for all positive integers q. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Preservation of the joint essential matricial range

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References (22)

Publisher
Wiley
Copyright
© 2019 London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms.12279
Publisher site
See Article on Publisher Site

Abstract

Let A=(A1,⋯,Am) be an m‐tuple of elements of a unital C∗‐algebra 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:A→Mq. When A=B(H), where B(H) is the algebra of bounded linear operators on the Hilbert space H, the joint spatial q‐matricial range Wsq(A) of A is the set of (B1,⋯,Bm)∈Mqm for which there is a q‐dimensional subspace V of H such that Bj is the compression of Aj to V for j=1,⋯,m. Suppose that K(H) is the set of compact operators in B(H). The joint essential spatial q‐matricial range is defined as Wessq(A)=∩{cl(Wsq(A1+K1,⋯,Am+Km)):K1,⋯,Km∈K(H)},where cl(T) denotes the closure of the set T. 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 clWsq(A1+K1,⋯,Am+Km)=Wessq(A)forallq∈{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(A) is a simplex in Rm, then we prove that there are self‐adjoint compact operators K1,⋯,Km such that cl(Wsq(A1+K1,⋯,Am+Km))=Wessq(A) for all positive integers q.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Oct 1, 2019

Keywords: ; ;

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