We unraveled the relation morphisms of graphs simultaneously generalizing both the path homomorphisms of graphs and the standard graph homomorphisms. Then we determined the admissibility conditions for relation morphisms that allowed us to define a functor to the category of algebras and algebra homomorphisms. This functor, when restricted to the subcategory of graphs and admissible path homomorphisms, reproduced the known covariant functor, and when restricted to the subcategory of graphs and admissible standard graph homomorphisms, reproduced the known contravariant functor. Finally, using admissible relations of graphs, we unified the mixed-pullback theorem and the pushout-to-pullback theorem into one pullback theorem. Better still, we found interesting applications of our new formalism in the study of unital embeddings of Cuntz algebras.
Groundbreaking new frameworks for obtaining classification goals have been established, including a surprising new characterization of the key concept of “shift equivalence”. The general classification problem requires new ideas, but results for equivariant classification problem surpass our expectations. Also, striking applications of the equivariant results to standard quantum spaces have been obtained. There is rapid progress in studying classification in the algebraic setting obtained by appealing to C*-algebraic results or methods, with no less promising results in the opposite direction. Better still, a certain classical operator-theoretic question was answered by analyzing higher-rank graph C*-algebras, and a very tight connection between the notions of sameness of the very general Ott-Tomforde-Willis models for non-compact shift spaces with C*-algebraic characterizations was unraveled.
For a class of graph algebras, we have shown that there is a group isomorphism between their K-groups and a quotient polynomial ring constructed from the adjacency matrix of the graph, and that this isomorphism can be promoted to a homomorphism of semirings. Concrete examples include the C*-algebra of the quantum projective space, the UHF algebra, and the C*-algebra of the space parameterizing Penrose tilings. For the quantum spheres, the graph C*-algebraic description dates back to a paper of Hong and Szymański, while a previous description using groupoids is due to Sheu. Furthermore, we have shown that the path groupoid of the directed graph of Hong and Szymański is isomorphic to the groupoid discovered by Sheu. Finally, we discovered that a correspondence itself can be generated from functors associated with a (quantum) graph, and that the category of modules over its Cuntz-Pimsner algebra is equivalent to a stable category of a canonical adjunction.
A full understanding of the situation concerning the quantum SU(2) has been obtained as well as a proof that the quantum projective spaces and higher-dimensional quantum spheres are indeed quantum metric spaces. Furthermore, the approximations that are only finite-dimensional relative to a C*-algebra have also been investigated.
Progress has been made regarding equivariant dimensions on continuous fields of C*-algebras. Furthermore, the relation between the finiteness of the local-triviality dimensions and the notion of freeness was examined in detail. Next, the study of coactions on matrix algebras, has been conducted. Graph C*-algebras are C*-algebras arising from left cancellative categories. Coactions of discrete groups on such algebras were investigated, and a principal quiver bundle was defined. This is the first step in studying equivariant dimensions of topological quiver C*-algebra, which will generalize the previously obtained results for graph C*-algebras.
Significant progress has been made in the construction of new equivariant spectral triples on Cuntz-Krieger algebras. By taking a dynamical-system point of view, the noncommutative geometry of Cuntz-Krieger algebras can be understood both topologically and metrically via an intrinsically defined operator called the Logarithmic Dirichlet Laplacian. A first main breakthrough was achieved by carrying out a fundamental analysis of this operator, in the full generality of Ahlfors regular metric measure spaces. Subsequently, this operator was shown to exist in the context of the dynamical groupoid underlying Cuntz-Krieger algebras, where it was used to define new spectral triples in a systematic way. These spectral triples exhaust the odd K-homology of Cuntz-Krieger algebras. Furthermore, we completely settled the question of equivariance of spectral triples, discovering unexpected new properties.
The main efforts were directed to unravelling the basics of the notion of isometry. Here, the main results include the definition of a generalisation of the metric and other tensors to the general framework of spectral noncommutative geometry that allows one to redefine the notion of isometries and study special symmetric examples of quantum spaces. In particular, it was shown that the geometry of the quantum SU(2) C*-algebra is torsion free. In a more general setting, a characterization of the families of bialgebras or Hopf algebras over fields for which the product in the corresponding category is finite-dimensional, was obtained. The result showed that, if the ground field is infinite, then bialgebra or Hopf-algebra products are finite-dimensional precisely when the factors are, with at most one of dimension bigger than 1. Further results concern properties of actions of compact quantum groups on Banach spaces, and work on the geometries of infinite binary graphs and symmetries of geometries of graphs is in progress.
