# Problems on exercise 7.G in the book “K-Theory and C*-Algebras”

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

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?

• Ad 1.) Indeed, the C*-algebra generated by two commuting unitaries is $C(\mathbb T^2)$, where $\mathbb T^2 = S^1 \times S^1$. – André S. Dec 1 '18 at 15:09
• @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. – Martin Argerami Dec 1 '18 at 17:00
• Right. So then I mean: Is a quotient of. – André S. Dec 2 '18 at 8:55