- Gelfand–Naimark theorem
In
mathematics , the Gelfand–Naimark theorem states that an arbitraryC*-algebra "A" is isometrically *-isomorphic to a C*-algebra ofbounded operator s on aHilbert space . This result was a significant point in the development of the theory of C*-algebras in the early1940 s since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an algebra of operators.The Gelfand–Naimark representation π is the
direct sum of representations π"f"of "A" where "f" ranges over the set ofpure state s of A and π"f" is the irreducible representation associated to "f" by theGNS construction . Thus the Gelfand–Naimark representation acts on the Hilbert direct sum of the Hilbert spaces "H""f" by:pi(x) [igoplus_{f} xi_f] = igoplus_{f} pi_f(x)xi_f.
Note that π("x") is a bounded linear operator since it is the direct sum of a family of operators, each one having norm ≤ ||"x"||.
Theorem. The Gelfand–Naimark representation of a C*-algebra is an isometric *-representation.
It suffices to show the map π is injective, since for *-morphisms of C*-algebras injective implies isometric. Let "x" be a non-zero element of "A". By the
Krein extension theorem for positive linear functionals, there is a state "f" on "A" such that "f"("z") ≥ 0 for all non-negative z in "A" and "f"(−"x"* "x") < 0. Consider the GNS representation π"f" with cyclic vector ξ. Since:pi_f(x) xi|^2 = langle pi_f(x) xi mid pi_f(x) xi angle = langle xi mid pi_f(x^*) pi_f(x) xi angle = langle xi mid pi_f(x^* x) xi angle= f(x^* x) > 0,
it follows that π"f" ≠ 0. Injectivity of π follows.
The construction of Gelfand–Naimark "representation" depends only on the
GNS construction and therefore it is meaningful for any B*-algebra "A" having anapproximate identity . In general it will not be afaithful representation . The closure of the image of π("A") will be a C*-algebra of operators called theC*-enveloping algebra of "A". Equivalently, we can define the C*-enveloping algebra as follows: Define a real valued function on "A" by:x|_{operatorname{C}^*} = sup_f sqrt{f(x^* x)} as "f" ranges over pure states of "A". This is a semi-norm, which we refer to as the "C* semi-norm" of "A". The set I of elements of "A" whose semi-norm is 0 forms a two sided-ideal in "A" closed under involution. Thus the quotient vector space "A" / I is an involutive algebra and the norm:cdot ||_{operatorname{C}^*}
factor s through a norm on "A" / I, which except for completeness, is a C* norm on "A" / I (these are sometimes called pre-C*-norms). Taking the completion of "A" / I relative to this pre-C*-norm produces a C*-algebra "B".By the
Krein-Milman theorem one can show without too much difficulty that for "x" an element of the B*-algebra "A" having an approximate identity::sup_{f in operatorname{State}(A)} f(x^*x) = sup_{f in operatorname{PureState}(A)} f(x^*x) It follows that an equivalent form for the C* norm on "A" is to take the above supremum over all states.The universal construction is also used to define
universal C*-algebra s of isometries.Remark. The
Gelfand representation orGelfand isomorphism for a commutative C*-algebra with unit A is an isometric *-isomorphism from A to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals of "A" with the weak* topology.References
*
I. M. Gelfand andM. A. Naimark , "On the imbedding of normed rings into the ring of operators on a Hilbert space", Math. Sbornik, vol 12, 1943. [http://www.google.com/books?id=DYCUp0JYU6sC&printsec=frontcover#PPA3,M1 Available online] throughGoogle Book Search .
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