State on a C*-algebra

Summary

A state on a C*-algebra is a positive linear functional of norm $1$. On a unital C*-algebra this means a linear functional that takes nonnegative values on positive elements and the value $1$ at the unit. States play the role of probability measures. For a compact Hausdorff space $X$, the states on $C(X)$ are the integrals against regular Borel probability measures on $X$. In quantum mechanics, the states of a system with observable algebra $A$ are the states on $A$.