On Projections in the Noncommutative 2-Torus Algebra

We investigate a set of functional equations defining a projection in the noncommutative 2-torus algebra $A_{\theta}$. The exact solutions of these provide various generalisations of the Powers-Rieffel projection. By identifying the corresponding $K_0(A_{\theta})$ classes we get an insight into the structure of projections in $A_{\theta}$.

💡 Research Summary

The paper investigates projections in the irrational rotation algebra (A_{\theta}), the C(^*)-algebra generated by two unitaries (U) and (V) satisfying (VU=e^{2\pi i\theta}UV) with irrational (\theta). After recalling the smooth dense subalgebra (A_{\theta}^{\infty}) and the canonical trace (\tau), the author writes a generic element of the form
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