Compressions of compact tuples Passer, Benjamin; Shalit, Orr Moshe
Linear algebra and its applications,
03/2019, Letnik:
564
Journal Article
Recenzirano
Odprti dostop
We study the matrix range of a tuple of compact operators on a Hilbert space and examine the notions of minimal, nonsingular, and fully compressed tuples. In this pursuit, we refine previous results ...by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range. Further, we find that a compact tuple A is fully compressed if and only if it is multiplicity-free and the Shilov ideal is trivial, which occurs if and only if A is minimal and nonsingular. Fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges. We also produce a proof of this fact which does not depend on the concept of nonsingularity.
The purpose of this paper is to give a self-contained overview of the theory of matrix convex sets and free spectrahedra. We will give new proofs and generalizations of key theorems. However we will ...also introduce various new concepts and results as well. Key contributions of this paper are
A new general Krein–Milman theorem that characterizes the smallest operator tuple defining a compact matrix convex set.
An introduction and a characterization of matrix exposed points.
A (weak) Minkowski theorem in the language of matrix extreme points (with a new proof of the weak Krein–Milman theorem of Webster and Winkler).
Simplified/new proofs of the Gleichstellensatz, Helton and McCullough’s characterization of free spectrahedra as closures of matrix convex “free basic open semialgebraic” sets and a characterization of hermitian irreducible free loci of Helton, Klep and Vol
c
ˇ
i
c
ˇ
.
A classic result of Namioka and Phelps on the extreme points of the state space on the tensor product of order unit spaces is generalized to the setting of matrix convexity. We show that the matrix ...extreme points of the matrix state space on the tensor product of two unital
C
∗
-algebras, at least one of them is of type
I
, are exactly the isometric orbit of the restrictions, on the exponentially unitary orbit of the
C
∗
-algebraic tensor product, of the tensor products of the matrix extreme points of matrix state spaces of the two
C
∗
-algebras.
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