Using a result of Vdovina, we may associate to each complete connected bipartite graph $\kappa $ a two-dimensional square complex, which we call a tile complex, whose link at each vertex is $\kappa $ . We regard the tile complex in two different ways, each having a different structure as a $2$ -rank graph. To each $2$ -rank graph is associated a universal $C^{\star }$ -algebra, for which we compute the K-theory, thus providing a new infinite collection of $2$ -rank graph algebras with explicit K-groups. We determine the homology of the tile complexes and give generalisations of the procedures to complexes and systems consisting of polygons with a higher number of sides.

We develop a theory of graph algebras over general fields. This is modelled after the theory developed by Freedman et al. (2007, J. Amer. Math. Soc.20 37–51) for connection matrices, in the study of graph homomorphism functions over real edge weight and positive vertex weight. We introduce connection tensors for graph properties. This notion naturally generalizes the concept of connection matrices. It is shown that counting perfect matchings, and a host of other graph properties naturally defined as Holant problems (edge models), cannot be expressed by graph homomorphism functions with both complex vertex and edge weights (or even from more general fields). Our necessary and sufficient condition in terms of connection tensors is a simple exponential rank bound. It shows that positive semidefiniteness is not needed in the more general setting.

Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$-algebras with additional $C^{*}$-algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.

We study the problem of extending a state on an abelian $C^{\ast }$-subalgebra to a tracial state on the ambient $C^{\ast }$-algebra. We propose an approach that is well suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient $C^{\ast }$-algebra; we prove that these extensions are tracial states if and only if certain invariance properties of both the state and conditional expectations are satisfied. In the example of a groupoid $C^{\ast }$-algebra, these invariance properties correspond to invariance of associated measures on the unit space under the action of bisections. Using our framework, we are able to completely describe the tracial state space of a Cuntz–Krieger graph algebra. Along the way we introduce certain operations called graph tightenings, which both streamline our description and provide connections to related finiteness questions in graph $C^{\ast }$-algebras. Our investigation has close connections with the so-called unique state extension property and its variants.

Let $\mathfrak{A}$ be a ${{C}^{*}}$-algebra with real rank zero that has the stable weak cancellation property. Let $\Im $ be an ideal of $\mathfrak{A}$ such that $\Im $ is stable and satisfies the corona factorization property. We prove that

$$0\,\to \,\Im \,\to \mathfrak{A}\,\to \,\mathfrak{A}/\Im \,\to \,0$$

is a full extension if and only if the extension is stenotic and $K$-lexicographic. As an immediate application, we extend the classification result for graph ${{C}^{*}}$-algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West, and the first named author, our result may also be used to give a purely $K$-theoretical description of when an essential extension of two simple and stable graph ${{C}^{*}}$-algebras is again a graph ${{C}^{*}}$- algebra.