Preservation of the joint essential matricial range

Abstract

Let A = ( A 1 , ⋯ , A m ) be an m-tuple of elements of a unital C ∗ -algebra A and let M q denote the set of q × q complex matrices. The joint q-matricial range W q ( A ) is the set of ( B 1 , ⋯ , B m ) ∈ M q m such that B j = Φ ( A j ) for some unital completely positive linear map Φ : A → M q . 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 W s q ( A ) of A is the set of ( B 1 , ⋯ , B m ) ∈ M q m for which there is a q-dimensional subspace V of H such that B j is the compression of A j 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 W e s s q ( A ) = ∩ { cl ( W s q ( A 1 + K 1 , ⋯ , A m + K m ) ) : K 1 , ⋯ , K m ∈ 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 W e s s q ( A ) = W q ( π ( A ) ) , where π ( A ) = ( π ( A 1 ) , ⋯ , π ( A m ) ) . Furthermore, for any positive integer N, we prove that there are self-adjoint compact operators K 1 , ⋯ , K m such that cl W s q ( A 1 + K 1 , ⋯ , A m + K m ) = W e s s q ( A ) 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 W e s s 1 ( A ) is a simplex in R m , then we prove that there are self-adjoint compact operators K 1 , ⋯ , K m such that cl ( W s q ( A 1 + K 1 , ⋯ , A m + K m ) ) = W e s s q ( A ) for all positive integers q.