---
tag: 4DSVM5Q17H
kind: concept
name: Operator system
msc: [46L07, 47L07]
sources:
  - url: https://en.wikipedia.org/wiki/Operator_system
same_as:
  wikidata: Q7097842
  wikipedia: Operator system
---

## Summary

An operator system is a subspace of a unital C*-algebra that contains the unit and is closed under taking adjoints.
Such a subspace has enough positive elements to make sense of positive and [[completely positive maps|CGY299QMC3]] on it.
Operator systems are the setting of the Arveson extension theorem and of much of the theory of completely positive maps.
A closed operator system is in particular an [[operator space|VB1J9AA4WX]].

## Explanations

### L1 human

Inside an algebra of operators, an operator system keeps just enough structure to talk about positivity.
It must contain the identity, and with each element it must contain the adjoint.
It need not be closed under multiplication.
For example, the functions $a + b z + c \bar z$ on the unit circle, with $a, b, c$ complex, form an operator system in the continuous functions on the circle.