Possibly non-unital operator system structures on a possibly non-unital function system

Abstract

In this paper, we first give the definition of possibly non-unital function system, which is a characterization of the self-adjoint subspace of the space of continuous functions on a compact Hausdorff space with the induced order and norm structure. Similar to operator system case, we define the unitalization of a possibly non-unital function system. Then we construct two possibly non-unital operator system structures on a given possibly non-unital function system, which are the analogues of minimal and maximal operator spaces on a normed space. These two structures have many interesting relations with the minimal and maximal operator system structures on a given function system.

Keywords

Possibly non-unital function system / operator system / possibly non-unital operator system

References

[1] Effros E. G., Ruan Z. -J. Operator Spaces, 2000, Oxford: Oxford Univ Press

[2] Kavruk A. S., Paulsen V. I., Todorov I. G., Tomforde A. M. Tensor products of operator systems. J Funct Anal, 2011, 261: 267-299

[3] Ng C K. Operator subspaces of L(H) with induced matrix orderings. Indiana Univ Math J (to appear)

[4] Paulsen V. I., Todorov I. G., Tomforde A. M. Operator system structures on ordered spaces. Proc Lond Math Soc (3), 2011, 102(1): 25-49

[5] Paulsen V. I., Tomforde A. M. Vector spaces with an order unit. Indiana Univ Math J, 2009, 58: 1319-1359

[6] Werner W. Subspaces of L(H) that are *-invariant. J Funct Anal, 2002, 193: 207-223

[7] Werner W. Multipliers on matrix ordered operator spaces and some K-groups. J Funct Anal, 2004, 206: 356-378