The class of $\lambda $-synchronizing subshifts generalizes the class of irreducible sofic shifts. A $\lambda $-synchronizing subshift can be presented by a certain $\lambda $-graph system, called the $\lambda $-synchronizing $\lambda $-graph system. The $\lambda $-synchronizing $\lambda $-graph system of a $\lambda $-synchronizing subshift can be regarded as an analogue of the Fischer cover of an irreducible sofic shift. We will study algebraic structure of the ${C}^{\ast } $-algebra associated with a $\lambda $-synchronizing $\lambda $-graph system and prove that the stable isomorphism class of the ${C}^{\ast } $-algebra with its Cartan subalgebra is invariant under flow equivalence of $\lambda $-synchronizing subshifts.

We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on separated normal surfaces any coherent sheaf is the quotient of a vector bundle. As a consequence, for such surfaces the Quillen K-theory of vector bundles coincides with the Waldhausen K-theory of perfect complexes. Examples show that, on non-separated schemes, usually many coherent sheaves are not quotients of vector bundles.

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.