For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${{\mathcal{A}}_{\theta }}$, and we show that the ${{C}^{*}}$ algebra $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ generated by these Toeplitz operators is an extension of ${{\mathcal{A}}_{\theta }}$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $K{{K}^{1}}\left( {{\mathcal{A}}_{\theta }},\mathbb{C} \right)$. We also prove an index theorem for $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ that generalizes the standard index theorem for Toeplitz operators on the circle.

For a dense $G_\delta$-set of parameters, the irrational rotation algebra is shown to contain infinitely many C*-subalgebras satisfying the following properties. Each subalgebra is isomorphic to a direct sum of two matrix algebras of the same (perfect square) dimension; the Fourier transform maps each summand onto the other; the corresponding unit projection is approximately central; the compressions of the canonical generators of the irrational rotation algebra are approximately contained in the subalgebra.

Let ${{A}_{\theta}}$ denote the rotation algebra—the universal ${{C}^{*}}$-algebra generated by unitaries $U,V$ satisfying $VU={{e}^{2\pi i\theta }}UV$, where $\theta $ is a fixed real number. Let $\sigma $ denote the Fourier automorphism of ${{A}_{\theta}}$ defined by $U\mapsto V\text{,}V\mapsto {{U}^{-1}}$, and let ${{B}_{\theta }}={{A}_{\theta }}{{\rtimes }_{\sigma }}\mathbb{Z}/4\mathbb{Z}$ denote the associated ${{C}^{*}}$-crossed product. It is shown that there is a canonical inclusion ${{\mathbb{Z}}^{9}}\hookrightarrow {{K}_{0}}({{B}_{\theta }})$ for each $\theta $ given by nine canonical modules. The unbounded trace functionals of ${{B}_{\theta }}$ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero $\text{H}{{\text{C}}^{0}}({{B}_{\theta }})$ when $\theta $ is irrational. The Chern characters of the nine modules—and more importantly, the Fourier module—are computed and shown to involve techniques from the theory of Jacobi’s theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense ${{\text{G}}_{\delta }}$ set of values of $\theta $ one has ${{K}_{0}}({{B}_{\theta }})\cong {{\mathbb{Z}}^{9}}$ and is generated by the nine classes constructed here.