Gelfand–Naimark theorem

In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra "A" is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. This result was a significant point in the development of the theory of C*-algebras in the early 1940s 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 of pure states of A and π_"f" is the irreducible representation associated to "f" by the GNS construction. Thus the Gelfand–Naimark representation acts on the Hilbert direct sum of the Hilbert spaces "H"_"f" by

:

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 an approximate identity. In general it will not be a faithful representation. The closure of the image of π("A") will be a C*-algebra of operators called the C*-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

:$||\; cdot\; ||\_\{operatorname\{C\}^*\}$

factors 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*-algebras of isometries.

Remark. The Gelfand representation or Gelfand 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 and M. 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] through Google Book Search.