SEMINARS
Preservation of the joint essential matricial range

2018-07-26　16:00 — 17:30

Middle Lecture Room

Yiu-Tung Poon

Iowa State University, USA

Let A = (A1; : : : ;Am) be an m-tuple of self-adjoint elements of a unital C*-algebra
A. The joint q-matricial range Wq(A) is the set of (B1; : : : ;Bm) 2 Mm
q such that Bj = (Aj)
for some unital completely positive linear map  : 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
Wq
s (A) ofAis the set of (B1; : : : ;Bm) 2 Mm
q such that Bj is a compression of Aj on a q-dimensional
subspace. The joint essential spatial q-matricial range is defined as
Wq
ess(A) = \fcl(Wq
s (A1 + K1; : : : ;Am + Km)) : K1; : : : ;Km are compact operatorsg;
where cl denotes the closure. Suppose K(H) is the set of compact operators in B(H), and  is
the canonical surjection from B(H) to the Calkin algebra B(H)=K(H). We prove that Wq
ess(A) is
C-convex and equals the joint q-matricial range Wq((A)), where (A) = ((A1); : : : ; (Am)).
Furthermore, for any positive integer N, we prove that there are self-adjoint compact operatorsK1; : : : ;Km such that
Wq
s (A1 + K1; : : : ;Am + Km) = Wq
ess(A) for all q 2 f1; : : : ;Ng:
If W1
ess(A) = W1((A)) is a simplex in Rm, then we prove that there are self-adjoint compact
operators K1; : : : ;Km such that Wq
s (A1 + K1; : : : ;Am + Km) = Wq
ess(A) for all positive integers q.
These results generalize those of Narcowich-Ward and Smith-Ward, obtained in the m = 2 case, and
also generalize the result of Müller when m  1 and q = 1.
This is joint work with Chi-Kwong Li and Vern I. Paulsen.