Positive functional

on an algebra $A$ with an involution ${} ^ {*}$

A linear functional $f$ on the ${} ^ {*}$- algebra $A$ that satisfies the condition $f( x ^ {*} x) \geq 0$ for all $x \in A$. Positive functionals are important and have been introduced in particular because they are used in the GNS-construction, which is one of the basic methods for examining the structures of Banach ${} ^ {*}$- algebras. This and its generalizations, for example to weights in $C ^ {*}$- algebras, provide the basis for proving the theorem on the abstract characterization of uniformly-closed ${} ^ {*}$- algebras of operators on a Hilbert space and the theorem on the completeness of a system of irreducible unitary representations of a locally compact group.

The GNS-construction is a method for constructing a ${} ^ {*}$- representation $\pi _ {f}$ of a ${} ^ {*}$- algebra $A$ with unit on a Hilbert space $H _ {f}$ for any positive functional $f$ on $A$, which is such that $f( x) = \langle \pi _ {f} ( x) \xi , \xi \rangle$ for all $x \in A$, where $\xi \in H _ {f}$ is a certain cyclic vector. The construction is the following: The semi-inner product $\langle x, y \rangle = f( y ^ {*} x)$ is defined on $A$; the corresponding neutral subspace is a left ideal $N _ {f} = \{ {x \in A } : {f( x ^ {*} x) = 0 } \}$, and therefore in the pre-Hilbert space $A / N _ {f}$ left-multiplication operators $L _ {a}$ by the elements $a \in A$( $L _ {a} ( x + N _ {f} ) = ax + N _ {f}$) are well-defined; the operators $L _ {a}$ are continuous and can be extended to continuous operators $\overline{L}\; _ {a}$ on the completion $H _ {f}$ of $A / N _ {f}$. The mapping $\pi _ {f}$ that takes $a \in A$ to $\overline{L}\; _ {a}$ is the required representation, where for $\xi$ one can take the image of the unit under the composition of the canonical mappings $A \rightarrow A / N _ {f} \rightarrow H _ {f}$.

References

[1]	I.M. Gel'fand, M.A. Naimark, "Normed involution rings and their representations" Izv. Akad. Nauk SSSR Ser. Mat. , 12 (1948) pp. 445–480 (In Russian)
[2]	I. Segal, "Irreducible representations of operator algebras" Bull. Amer. Math. Soc. , 53 (1947) pp. 73–88
[3]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)