- -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.
- -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''.
- -ALGEBRA
- -algebra" class="copylinks">Spectrum Of A C
algebra ''
- -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 s, 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'').
- -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
- compact.
- -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
- convergence. In the general case, removal of a single point from a compact Hausdorff space yields a locally compact Hausdorff space.
- -algebra and let ''X'' be the spectrum of ''A''. The , or the '''Gelfand representation''' γ on ''A'' is defined as follows:
:
- -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''.
- -homomorphism and that the Spectral Radius of ''x'' equals the norm of ''x''. See the Arveson reference below.
- -algebra can also be viewed as the set of all Maximal Ideal s ''m'' of ''A'', with the Hull-kernel Topology . For any such ''m'' it is shown that ''A/m'' is Natural ly identified to the field of Complex Number s ''C''. Therefore any ''a'' in ''A'' gives rise to a complex-valued function on ''Y''.
- -algebras and morphisms into the category of locally compact Hausdorff spaces and continuous maps.
- -algebras ''A'', which though not quite analogous to the Gelfand representation, does provide a concrete representation of ''A'' as an algebra of operators.
- -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:
- -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.
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.
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