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I have a lot problems on exercise 7.G in the book K-Theory and C*-Algebras by Wegge-Olsen. enter image description here $\newcommand{\C}{\mathbb{C}}$

  1. $X\subset \mathbb{C}$? As I know the character space of $C^*(u_1,u_2)$ is homeomorphic to a subset of $\C^2$: $\{(\tau(u_1),\tau(u_2))|\tau \mbox{ is a character of } C^*(u_1,u_2)\}$.

  2. The example for a standard unitary should be $t\mapsto \exp(\frac{2\pi it}{1+|t|})$ as my tutor points out.

  3. When $A$ is unital, $(SA)^\sim=\{f\in C(\mathbb{T}\to A)|f(1)\in\C\}$. Why does $u_1:=1\otimes u \in M_n((SA)^\sim)$?

  4. Most important, what does the author want to tell us?

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  • $\begingroup$ Ad 1.) Indeed, the C*-algebra generated by two commuting unitaries is $C(\mathbb T^2)$, where $\mathbb T^2 = S^1 \times S^1$. $\endgroup$ – user42761 Dec 1 '18 at 15:09
  • $\begingroup$ @AndréS.: sometimes, but not always. If you take $u_1=u_2$, you will get $\mathbb T$ or a subset of it. And even if $u_1$ and $U_2$ are free, they may still have discrete spectrum. $\endgroup$ – Martin Argerami Dec 1 '18 at 17:00
  • $\begingroup$ Right. So then I mean: Is a quotient of. $\endgroup$ – user42761 Dec 2 '18 at 8:55

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