Gelfand–Naimark–Segal construction

In functional analysis, a discipline within mathematics, given a ${\displaystyle C^{*}}$-algebra ${\displaystyle A}$, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic ${\displaystyle *}$-representations of ${\displaystyle A}$ and certain linear functionals on ${\displaystyle A}$ (called states). The correspondence is shown by an explicit construction of the ${\displaystyle *}$-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.