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I am trying to solve the following exercise from "an introduction to K-theory" by M. Rørdam:

For every $C^*$-algebra A put $\mathbb{T}A=C(\mathbb{T},A)$, where $\mathbb{T}= \lbrace z \in \mathbb{C}: \vert z \vert =1 \rbrace$

(i) Construct a split exact sequence

$$0 \longrightarrow SA \longrightarrow \mathbb{T}A \leftrightarrows A \longrightarrow 0$$

For this part I have by definition that I am to construct a split exact sequence

$$0 \longrightarrow C_0((0,1),A) \longrightarrow C(\mathbb{T},A) \leftrightarrows A \longrightarrow 0$$

As SA is the suspension of A ($SA=\lbrace f \in C([0,1],A): f(0)=f(1)=0 \rbrace$ ). I can't seem to find the right maps, and I have also tried to make an isomorphism from $C(\mathbb{T},A)$ to $SA \oplus A$ as (I think) this would make it a split exact sequence as well and also it would be nice for the next part.

Is there another approach for this?

(ii) Show that $K_n(\mathbb{T}A)$ is isomorphic to $K_n(A) \oplus K_{n+1} (A)$ for every positive integer n.

I belive I can make the following:

\begin{align*} K_n(A) \oplus K_{n+1}(A) &= K_0(S^nA) \oplus K_0(S^{n+1}A) \\ &\cong K_0(S^nA \oplus S^{n+1}A) \\ &= K_0(S^nA \oplus S^n(SA)) \\ &\cong K_0(S^n (A \oplus SA) ) \\ &\cong K_0(S^n(\mathbb{T} A )) \\ &= K_n (\mathbb{T} A) \end{align*}

Is this true?

(iii) Show that $\mathbb{T}^n \mathbb{C}$ is isomorphic to $C(\mathbb{T}^n)$ and use this and (ii) to express $K_0(C(\mathbb{T}^n))$ and $K_1(C(\mathbb{T}^n))$ in terms of the groups $K_m(\mathbb{C})$. (only for n=1,2,3)

For this part I can not see how to get started so is there a hint for this?

Miep
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1 Answers1

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For (i), define the map $\psi:C(\mathbb T,A)\to A$ by $\psi(f)=f(1)$ (where $\mathbb T=\{z\in\mathbb C:|z|=1\}$), and define $s:A\to C(\mathbb T,A)$ by taking $a\in A$ to the constant function: $(sa)(z)=a$ for all $z\in A$. Show that $\psi\circ s=\operatorname{id}_A$, and that $\ker(\psi)\cong SA$.

For (ii), this follows from two facts in $K$-theory: A split exact sequence $0 \longrightarrow I \longrightarrow A \leftrightarrows B \longrightarrow 0$ of $C^*$-algebras induces split exact sequences $0 \longrightarrow K_i(I) \longrightarrow K_i(A) \leftrightarrows K_i(B) \longrightarrow 0$ in $K$-groups, and $K_i(SA)\cong K_{i+1}(A)$ for any $C^*$-algebra $A$.

For (iii), Show that $XYA=(X\times Y)A$ for any compact spaces $X,Y$ and $C^*$-algebra $A$, then proceed by induction. For the next part, begin with $n=1$, apply (ii) with $A=\mathbb C$ to compute $K_i(C(\mathbb T))$, then move to $n=2$ and apply (ii) with $A=C(\mathbb T)$.

Aweygan
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