Connected Unitaries in $K_{1}(C(\mathbb{T}^{3}))$

Question

Let $\mathbb{T}=\{\xi:|\xi|=1\}$ and set $A:=C(\mathbb{T}^{3})$. Then, $A$ is a unital commutative C$^{*}$-algebra. For any C$^{*}$-algebra $B$, let $\mathcal{U}(B)$ denote the unitary elements in $B$. For each $n$, set $\mathcal{U}_{n}(A):=\mathcal{U}(M_{n}(A))$, and put $\mathcal{U}_{\infty}(A)=\bigcup_{n\geq 1}\mathcal{U}_{n}(A)$. Then, the $K_{1}$ group of $A$ is defined to be $$ K_{1}(A):=\{[u]_{1}:u\in \mathcal{U}_{\infty}(A)\}. $$

Here $[\ \cdot \ ]_{1}$ denotes the equivalence class of the relation $\sim_{1}$ defined on $\mathcal{U}_{\infty}(A)$ as follows: for $u\in\mathcal{U}_{n}(A)$ and $v\in\mathcal{U}_{m}(A)$, $u\sim_{1} v$ if and only if there is a $k\geq\max\{n,m\}$ such that $u\oplus 1_{k-n}\sim_{h} v\oplus 1_{k-m}$ in $\mathcal{U}_{k}(A)$.

I know that $K_{1}(A)\cong \mathbb{Z}^{4}$ and that $\mathcal{U}(A)/\mathcal{U}_{0}(A)\cong \mathbb{Z}^{3}$, where $\mathcal{U}_{0}(A):=\{u\in\mathcal{U}:u\sim_{h} 1\}$. Therefore, there must be some $n\geq 2$ and some $u\in\mathcal{U}_{n}(A)$ such that for all $v\in\mathcal{U}(A)$, $[u]_{1}\not=[v]_{1}$.

However, in Rordam's C$^{*}$-algebra book, on page 201, he states that a more refined argument can show that $n=2$. I'm not sure why this is the case. I.e.,

I need to show that there is some $u\in\mathcal{U}_{2}(C(\mathbb{T}^{3}))$ such that for all $v\in\mathcal{U}(C(\mathbb{T}^{3}))$, $[u]_{1}\not=[v]_{1}$.

Answer 1

I don't know if there is another way to do this, but here is one proof. Recall that $K_1(A)$ is defined as the limit of the groups $$ \pi_0(GL_1(A)) \to \pi_0(GL_2(A))\to \pi_0(GL_3(A))\to \ldots $$ where the maps are induced by the natural map $GL_{n-1}(A)\to GL_n(A)$ given by $$ a\mapsto \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} $$ The question you ask would be answered if we knew when this sequence of groups stabilized. ie. Does there exist $n\in \mathbb{N}$ such that $$ K_1(A)\cong \pi_0(GL_{m-1}(A))\cong \pi_0(GL_m(A)) \quad\forall m\geq n\qquad (\dagger) $$ This is answered by Rieffel in this paper. Specifically, he defines two numbers associated to a $C^{\ast}$-algebra $A$

• Connected stable rank csr(A)
• General stable rank gsr(A)

He shows (Theorem 2.9 in the paper) that if $n=\max\{csr(A),gsr(C(\mathbb{T},A))\}$, then $(\dagger)$ holds.

Now, if $A = C(\mathbb{T}^3)$, then one can show (See, for instance, this paper by Nica)

• $csr(A) = 3$
• $gsr(C(\mathbb{T},A)) = gsr(C(\mathbb{T}^4)) = 1$

Hence, $(\dagger)$ holds for $n=3$, so that $$ K_1(C(\mathbb{T}^3)) \cong \pi_0(GL_2(C(\mathbb{T}^3)) $$ The result now follows from what you have mentioned.