I have seen it stated that for an open subset Y ⊆ X such that X is a compact Hausdorff space we get an identification of the C ∗ -algebras : C ( X ∖ Y ) ≅ C ( X ) / C 0 ( Y ).I suppose that this relies on the Tietze extension theorem but I fail to connect the dots.How do I realize this?

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I have seen it stated that for an open subset   Y  ⊆  X such that   X is a compact Hausdorff space we get an identification of the       C    ∗  -algebras :   C  (  X  ∖  Y  )  ≅  C  (  X  )      /        C    0    (  Y  ).I suppose that this relies on the Tietze extension theorem but I fail to connect the dots.How do I realize this?

Conor Frederick · Accepted Answer

Because   Y is open,   X∖  Y is compact. Tietze&#039;s extension theorem applies and given   f  ∈  C  (  X  ∖  Y  ) there exists             f      ~        ∈  C  (  X  ) with             f      ~                  |              X      ∖      Y        =  f and   ‖            f      ~        ‖  =  ‖  f  ‖.The natural thing seems to define a   ∗-epimorphism   C  (  X  )  →  C  (  X  ∖  Y  ) such that its kernel is       C    0    (  Y  ). That is we consider   γ  :  C  (  X  )  →  C  (  X  ∖  Y  ) to be the restriction map. Tietze&#039;s Extension Theorem, as mentioned above, guarantees that   γ is surjective. Is is straightforward that it is a   ∗-homomorphism.Finally, if   γ  (  f  )  =  0, then   f            |              X      ∖      Y        =  0. So   f is continuous on   Y and takes the value 0 on   ∂  Y. This means that the restriction of   f to   Y is in       C    0    (  Y  ); indeed, given   ε  &amp;gt;  0, the open set   V  =      |    f            |              −      1        [  0  ,  ε  ) is open and contains   X∖  Y, and its complement is a compact subset of   Y such that       |    f      |    &amp;lt;  ε outside of it.

I have seen it stated that for an open subset Y ⊆ X such that X i

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