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Gelfand representation
*** Shopping-Tip: Gelfand representation
In
mathematics, the '''Gelfand representation''' in
functional analysis allows a complete characterisation of commutative
C*-algebras as algebras of continuous complex-valued functions. The Gelfand representation theorem is one avenue in the development of
spectral theory for
normal operators.
The model algebra
For any
locally compact space locally compact Hausdorff space Hausdorff topological space ''X'', the space ''C''
0(''X'') of continuous complex-valued functions on ''X'' which
vanish at infinity is in a
natural way a commutative C*-algebra:
* The structure of algebra over the complex numbers is obtained by considering the pointwise operations of addition and multiplication.
* The involution is pointwise complex conjugation.
* The norm is the
uniform norm on functions.
Conversely given a commutative C*-algebra ''A'', one can produce a locally compact Hausdorff space ''X'' so that ''A'' is *-isomorphic to ''C''
0(''X''). Moreover, if ''A'' is
unital, then ''X'' is
compact, so ''C''
0(''X'') is equal to ''C''(''X''), the algebra of all continuous complex-valued functions on ''X''.
In fact the space ''X'' can be described precisely: it is the so-called spectrum of ''A''.
The spectrum of a commutative C*-algebra
:''See also:
Spectrum of a C*-algebra''
The '''spectrum''' or '''Gelfand space''' of a commutative C*-algebra ''A'', denoted ''Â'', consists of the set of ''non-zero'' complex-valued *-homomorphisms on ''A''. Elements of the spectrum are called '''characters''' on ''A''. The spectrum is a subset of the unit ball of ''A*'' and as such can be given the weak-* topology. In terms of convergence of
net_(mathematics) nets, this topology can be described as follows: a net {''f''
''k''}
''k'' of elements of the spectrum of ''A'' converges to ''f'' iff for each ''x'' in ''A'', the net of complex numbers {''f''
''k''(''x'')}
''k'' converges to ''f''(''x'').
Note that if ''A'' is a
separable C*-algebra, the weak-* topology is
metrizable. Thus the spectrum of a separable commutative C*-algebra ''A'' can be regarded as a metric space.
Note that ''spectrum'' is an overloaded word. It also refers to the spectrum σ(''x'') of an element ''x'' of an algebra with unit, that is the set of complex numbers ''r'' for which ''x'' - ''r'' 1 is not invertible in ''A''. The two notions are connected in the following way: σ(''x'') is the set of complex numbers ''f''(''x'') where ''f'' ranges over Gelfand space of ''A''. Equivalently, σ(''x'') is the
range of γ(''x''), where γ is the Gelfand representation defined below.
The
Banach-Alaoglu theorem of
functional analysis asserts that the
unit ball of the dual of a Banach space is weak-* compact.
It follows from the Banach-Alaoglu theorem that the spectrum of a commutative C*-algebra is a locally compact Hausdorff space. In the case the C*-algebra has a multiplicative unit element it is easy to see that the spectrum is actually ''compact'', since
the condition for a linear functional to be in the spectrum is closed under weak-* convergence. In the general case, removal of a single point from a compact Hausdorff space yields a locally compact Hausdorff space.
Statement of the theorem
Let ''A'' be a commutative C*-algebra and let ''X'' be the spectrum of ''A''. The '''Gelfand map''', or the '''Gelfand representation''' γ on ''A'' is defined as follows:
:
'''Theorem'''. The Gelfand map γ is an isometric *-isomorphism from ''A'' onto ''C''
0(''X'').
The idea of the proof is as follows. If ''A'' has an identity element, we claim that for any element ''x'' of ''A'', the range of values of the function γ(''x'') is the same as the spectrum of the element of ''x''. In fact λ is a spectral value of ''x'' iff ''x'' - λ 1 is not invertible iff
''x'' − λ 1 belongs to at least one maximal ideal ''m'' of ''A''.
Now by the
Gelfand-Mazur theorem on Banach fields, the quotient ''A/m'' is naturally identified with the
complex numbers ''C''. It remains to show the resulting homomorphism is a *-homomorphism and that the
spectral radius of ''x'' equals the norm of ''x''. See the Arveson reference below.
The spectrum of a commutative C*-algebra can also be viewed as the set of all
maximal ideals ''m'' of ''A'', with the
hull-kernel topology. For any such ''m'' it is shown that ''A/m'' is
natural transformation naturally identified to the field of
complex numbers ''C''. Therefore any ''a'' in ''A'' gives rise to a complex-valued function on ''Y''.
The Gelfand map give rise to a
contravariant functor from the category of C*-algebras and morphisms into the category of locally compact Hausdorff spaces and continuous maps.
The
Gelfand–Naimark theorem is a result for arbitrary (abstract)
commutative noncommutative C*-algebras ''A'', which though not quite analogous to the Gelfand representation, does provide a concrete representation of ''A'' as an algebra of operators.
Applications
One of the most significant applications is the existence of a continuous ''functional calculus'' for normal elements in C*-algebra ''A'': An element ''x'' is normal iff ''x'' commutes with its adjoint ''x*'', or equivalently iff it generates a commutative C*-algebra C*(''x''). By the Gelfand isomorphism applied to C*(''x'') this is *-isomorphic to an algebra of continuous functions on a locally compact space. This observation leads almost immediately to:
'''Theorem'''. Let ''A'' be a C*-algebra with identity and ''x'' an element of ''A''. Then there is a *-morphism ''f'' → ''f''(''x'') from the algebra of continuous functions on the spectrum σ(''x'') into ''A'' such that
* It maps 1 to the multiplicative identity of ''A'';
* It maps the identity function on the spectrum to ''x''.
This allows us to apply continuous functions to bounded normal operators on Hilbert space.
Commutative Banach algebras
The same construction may be carried out for a commutative
Banach algebra ''A''. In this case, the representation one obtains is a continuous homomorphism into ''C''
0(''X''), but it is not in general an isomorphism of Banach algebras.
Reference
*
William Arveson, ''An Invitation to C*-Algebra'', Springer-Verlag, 1981.
Category:Functional analysis
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