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)