spectral invariance theorem (for -algebras)
For -algebras (http://planetmath.org/CAlgebra) the situation is quite .
The spectral invariance theorem is a straightforward corollary of the next more general theorem about invertible elements in -subalgebras.
Theorem - Let be as above. Then is invertible in if and only if invertible in .
If is invertible in then it is clearly invertible in .
If is invertible in , then so is . Thus, .
Since is self-adjoint (http://planetmath.org/InvolutaryRing), (see this entry (http://planetmath.org/SpecialElementsInACAlgebraAndTheirSpectralProperties)), and so has no bounded (http://planetmath.org/Bounded) connected components.
By the spectral permanence theorem (http://planetmath.org/SpectralPermanenceTheorem) we must have . Hence, , i.e. is invertible in .
It follows that , i.e. is invertible in .
|Title||spectral invariance theorem (for -algebras)|
|Date of creation||2013-03-22 17:29:53|
|Last modified on||2013-03-22 17:29:53|
|Last modified by||asteroid (17536)|
|Synonym||spectral invariance theorem|
|Synonym||invariance of the spectrum of -subalgebras|
|Defines||invertibility in -subalgebras|