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An $n$-dimensional NC torus algebra $A_\theta^{(n)}$ is defined for any antisymmetric $n\times n$ matrix $\theta$ of real numbers as the universal $C^*$-algebra, generated by unitaries $U_1,\dots,U_n$, subject to relations $U_lU_j=e^{2\pi i \theta_{lj}} U_j U_l$. Particularly, for $n=3$ one has to specify 3 numbers $\theta_{12}, \theta_{23}, \theta_{13}$. The family of NC-tori algebras is one of the main toy-models in noncommutative geometry.

It is well-known, that $K_1(A_\theta^{(n)})\simeq \mathbb{Z}^{2^{n-1}}$.

I've succeeded (given $\theta_{12}$ and $\theta_{13}$ are both non-integer) writing an explicit formula for unitary $a\in A_\theta^{(3)}$, s.t. $[U_1]_1, [U_2]_1, [U_3]_1, [a]_1$ generate $K_1(A_\theta^{(3)})$ as an Abelian group. [The current proof I have works for at least one of $\theta_{12},\theta_{23},\theta_{13}$ being irrational, but I believe that it is not hard to construct a continuity argument to cover the rational case. This formula also provides ${n \choose 3}$ out of $2^{n-1}$ generators of $K_1(A_\theta^{(n)})$ for higher values of $n$, and another $n$ generators are trivial $[U_1]_1,\dots,[U_n]_1$. Also note, that except for the totally commutative case, when all $\theta_{kl}\in\mathbb{Z}$, we can change the algebra generators $U_1,U_2,U_3$ making both $\theta_{12}$ and $\theta_{13}$ non-integer. So that construction (with $n=3$) works (up to a change of generators) always except for the totally commutative case.]

Is this result (explicit formula for the 4-th generator) already written somewhere in the literature?

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