Hardy space

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 proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra.

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

$\pi (x)[\bigoplus _{f}\xi _{f}]=\bigoplus _{f}\pi _{f}(x)\xi _{f}.$

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

$\|\pi _{f}(x)\xi \|^{2}=\langle \pi _{f}(x)\xi \mid \pi _{f}(x)\xi \rangle =\langle \xi \mid \pi _{f}(x^{*})\pi _{f}(x)\xi \rangle =\langle \xi \mid \pi _{f}(x^{*}x)\xi \rangle =f(x^{*}x)>0,$

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 Banach *-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 Banach *-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, which in the commutative case are precisely the pure states, of A with the weak* topology.

References

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