Just a supplement to the answer by Tobias Fritz: All your examples are obviously commutative, since there is only one generator which is normal. Thus the question is really about finding certain terminal compact Hausdorff spaces. For example 1. comes from the terminal compact Hausdorff space $X$ equipped with a continuous function $X \to \mathbb{C}$ which is self-adjoint and norm $1$, i.e. whose image equals $[-1,1]$. This is obviously $[-1,1]$, equipped with the identity. You get the same answer when the norm is supposed to be $\leq 1$ (but $<1$ doesn't work). In a similar way one gets the other answers mentioned by Tobias Fritz.
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All your examples are obviously commutative, since there is only one generator which is normal. Thus the question is really about finding certain terminal compact Hausdorff spaces. For example 1. comes from the terminal compact Hausdorff space $X$ equipped with a continuous function $X \to \mathbb{C}$ which is self-adjoint and norm $1$, i.e. whose image equals $[-1,1]$. This is obviously $[-1,1]$, equipped with the identity. You get the same answer when the norm is supposed to be $\leq 1$ (but $<1$ doesn't work). In a similar way one gets the other answers mentioned by Tobias Fritz. |
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