TY - JOUR

T1 - Inductive limits in the operator system and related categories

AU - Mawhinney, Linda

AU - Todorov, Ivan G.

PY - 2019/1/10

Y1 - 2019/1/10

N2 - We present a systematic development of inductive limits in
the categories of ordered *-vector spaces, Archimedean order unit spaces,
matrix ordered spaces, operator systems and operator C*-systems. We
show that the inductive limit intertwines the operation of passing to
the maximal operator system structure of an Archimedean order unit
space, and that the same holds true for the minimal operator system
structure if the connecting maps are complete order embeddings. We
prove that the inductive limit commutes with the operation of taking
the maximal tensor product with another operator system, and establish
analogous results for injective functorial tensor products provided
the connecting maps are complete order embeddings. We identify the
inductive limit of quotient operator systems as a quotient of the inductive
limit, in case the involved kernels satisfy a lifting condition, implied
by complete biproximinality. We describe the inductive limit of graph
operator systems as operator systems of topological graphs, show that
two such operator systems are completely order isomorphic if and only
if their underlying graphs are isomorphic, identify the C*-envelope of
such an operator system, and prove a version of Glimm’s Theorem on
the isomorphism of UHF algebras in the category of operator systems.

AB - We present a systematic development of inductive limits in
the categories of ordered *-vector spaces, Archimedean order unit spaces,
matrix ordered spaces, operator systems and operator C*-systems. We
show that the inductive limit intertwines the operation of passing to
the maximal operator system structure of an Archimedean order unit
space, and that the same holds true for the minimal operator system
structure if the connecting maps are complete order embeddings. We
prove that the inductive limit commutes with the operation of taking
the maximal tensor product with another operator system, and establish
analogous results for injective functorial tensor products provided
the connecting maps are complete order embeddings. We identify the
inductive limit of quotient operator systems as a quotient of the inductive
limit, in case the involved kernels satisfy a lifting condition, implied
by complete biproximinality. We describe the inductive limit of graph
operator systems as operator systems of topological graphs, show that
two such operator systems are completely order isomorphic if and only
if their underlying graphs are isomorphic, identify the C*-envelope of
such an operator system, and prove a version of Glimm’s Theorem on
the isomorphism of UHF algebras in the category of operator systems.

U2 - 10.4064/dm771-4-2018

DO - 10.4064/dm771-4-2018

M3 - Article

VL - 536

JO - Dissertationes Mathematicae

JF - Dissertationes Mathematicae

SN - 0012-3862

ER -