1 GNS Construction
The Gelfand-Naimark-Segal construction (or GNS construction) is a fundamental idea in the of . It provides a procedure to construct and study representations of -algebras (http://planetmath.org/CAlgebra) and is the first step on the proof of the Gelfand-Naimark representation theorem, which that every -algebra is isometrically isomorphic to a closed *-subalgebra of , the algebra of bounded operators on a Hilbert space .
2 Representations associated with positive linear functionals
Let be a -algebra and a positive linear functional in .
We are going to construct a representation of and for that we need to construct a suitable Hilbert space.
Let us endow with a semi-inner product defined by . Now we define the set
It is easily seen that is a closed left ideal (http://planetmath.org/Ideal) in (using the Cauchy-Schwarz inequality, which is valid in semi-inner product spaces), so that induces a well defined inner product on the quotient (http://planetmath.org/QuotientModule) . The completion of is then an Hilbert space, which we will be denoted by .
Being bounded, the operator extends uniquely to a bounded operator on , which we denote by the same symbol, .
Let be the algebra of bounded operators on .
Theorem 2 - The function defined by is a -algebra representation of .
This representation is called the GNS representation associated to .
3 Cyclic vectors and GNS pairs
Suppose had an identity element . In this case it is easily seen that there exists a cyclic vector , i.e. a vector such that is dense (http://planetmath.org/Dense) in . This cyclic vector can just be chosen as .
Moreoever, this cyclic vector is such that for every .
Thus, in this case the representation is cyclic (http://planetmath.org/BanachAlgebraRepresentation) and is a vector state of . The result is still valid for general -algebras:
Theorem 3 - Let be the representation of defined previously. Then there exists a vector such that
is dense in , i.e. is cyclic,
for every , i.e. is a vector state.
Any pair , where is a representation of on a Hilbert space and , satisfying the above conditions for :
is dense in ,
is called a GNS pair for .
Theorem 4 - All GNS pairs for are (in the sense that the corresponding representations are unitarily equivalent).
4 Irreducible representations
We know that are ”plenty” of states on -algebra (see this entry (http://planetmath.org/PropertiesOfStates)), and so we have assured the existence of many (cyclic) representations. An interesting fact is that this representations associated to states are irreducible (http://planetmath.org/BanachAlgebraRepresentation) exactly when the state is a pure state:
Theorem 5 - Let be a state on . Then the representation is irreducible if and only if is a pure state.
Theorem 6 - Let be a -algebra. For every element there exists an irreducible representation of such that .
This last theorem is a fundamental step in the proof of the Gelfand-Naimark representation theorem.
|Date of creation||2013-03-22 17:47:40|
|Last modified on||2013-03-22 17:47:40|
|Last modified by||asteroid (17536)|
|Defines||pure states and irreducible representations|