- #1
Oxymoron
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Although this problem is meant to be easy I can't quite work it out.
Let [itex]U(A)[/itex] denote the set of unitary elements of a C*-algebra A. I've already shown that if u is unitary in A then the spectrum of u:
[tex]\sigma(u) \subset \mathbb{T} = \{z\in\mathbb{C}\,:\,|z|=1\}[/tex]
which was easy.
Now, apparently I can deduce that there exists a *-homomorphism [itex]\phi\,:\,C(\mathbb{T})\rightarrow A[/itex] from the compact space of continuous operators on the spectrum to the C*-algebra, such that [itex]\phi(\iota)\mathbb{C} = u[/itex]. Where [itex]\iota\,:\,\mathbb{T}\rightarrow\mathbb{C}[/itex] is the function defined by [itex]\iota(z) := z[/itex].
Now, i figured that the obvious choice for this *-homomorphism is the exponential function [itex]e^{it}[/itex] - but I could be wrong. However, if I am right, how should I go about proving that this is the correct deduction according to the question asked. I am assuming I will have to use the continuous functional calculus theorem somewhere.
Let [itex]U(A)[/itex] denote the set of unitary elements of a C*-algebra A. I've already shown that if u is unitary in A then the spectrum of u:
[tex]\sigma(u) \subset \mathbb{T} = \{z\in\mathbb{C}\,:\,|z|=1\}[/tex]
which was easy.
Now, apparently I can deduce that there exists a *-homomorphism [itex]\phi\,:\,C(\mathbb{T})\rightarrow A[/itex] from the compact space of continuous operators on the spectrum to the C*-algebra, such that [itex]\phi(\iota)\mathbb{C} = u[/itex]. Where [itex]\iota\,:\,\mathbb{T}\rightarrow\mathbb{C}[/itex] is the function defined by [itex]\iota(z) := z[/itex].
Now, i figured that the obvious choice for this *-homomorphism is the exponential function [itex]e^{it}[/itex] - but I could be wrong. However, if I am right, how should I go about proving that this is the correct deduction according to the question asked. I am assuming I will have to use the continuous functional calculus theorem somewhere.
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