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- In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra of bounded linear operators on some Hilbert space This article describes the spectral theory of closed normal subalgebras of (en)
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- 16438 (xsd:nonNegativeInteger)
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- Suppose is a closed normal subalgebra of that contains the identity operator and let be the maximal ideal space of Let be the Borel subsets of For every in let denote the Gelfand transform of so that is an injective map There exists a unique resolution of identity that satisfies:
the notation is used to summarize this situation.
Let be the inverse of the Gelfand transform where can be canonically identified as a subspace of Let be the closure of the linear span of
Then the following are true:
is a closed subalgebra of containing
There exists a isometric *-isomorphism extending such that for all
* Recall that the notation means that for all ;
* Note in particular that for all
* Explicitly, satisfies and for every .
If is open and nonempty then
A bounded linear operator commutes with every element of if and only if it commutes with every element of (en)
- Let be a resolution of identity on There exists a closed normal subalgebra of and an isometric *-isomorphism satisfying the following properties:
for all and which justifies the notation ;
for all and ;
an operator commutes with every element of if and only if it commutes with every element of
if is a simple function equal to where is a partition of and the are complex numbers, then ;
if is the limit of a sequence of simple functions in then converges to in and ;
for every (en)
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- In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra of bounded linear operators on some Hilbert space This article describes the spectral theory of closed normal subalgebras of (en)
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- Spectral theory of normal C*-algebras (en)
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