Piotr M. Hajac’s research while affiliated with Polish Academy of Sciences and other places

Associating graph algebras to directed graphs leads to both covariant and contravariant functors from suitable categories of graphs to the category k-Alg of algebras and algebra homomorphims. As both functors are often used at the same time, one needs a new category of graphs that allows a "common denominator" functor unifying the covariant and contravariant constructions. Herein, we solve this problem by first introducing the relation category of graphs RG, and then determining the concept of admissible graph relations that yields a subcategory of RG admitting a contravariant functor to k-Alg simultaneously generalizing the aforementioned covariant and contravariant functors. Although we focus on Leavitt path algebras and graph C*-algebras, on the way we unravel functors given by path algebras, Cohn path algebras and Toeplitz graph C*-algebras from suitable subcategories of RG to k-Alg. Better still, we illustrate relation morphisms of graphs by naturally occuring examples, including Cuntz algebras, quantum spheres and quantum balls.

Cuntz algebras $\mathcal{O}_n$, $n>1$, are celebrated examples of a separable infinite simple C*-algebra with a number of fascinating properties. Their K-theory allows an embedding of $\mathcal O_m$ in $\mathcal O_n$ whenever $n-1$ divides $m-1$. In 2009, Kawamura provided a simple and explicit formula for all such embeddings. His formulas can be easily deduced by viewing Cuntz algebras as graph C*-algebras. Our main result is that, using both the covariant and contravariant functoriality of assigning graph C*-algebras to directed graphs, we can provide explicit polynomial formulas for all unital embeddings of Cuntz algebras into matrices over Cuntz algebras allowed by K-theory.

In the standard category of directed graphs, graph morphisms map edges to edges. By allowing graph morphisms to map edges to finite paths (path homomorphisms of graphs), we obtain an ambient category in which we determine subcategories enjoying covariant functors to categories of algebras given by constructions of path algebras, Cohn path algebras, and Leavitt path algebras, respectively. Thus, we obtain new tools to unravel homomorphisms between Leavitt path algebras and between graph C*‐algebras. In particular, a graph‐algebraic presentation of the inclusion of the C*‐algebra of a quantum real projective plane into the Toeplitz algebra allows us to determine a quantum CW‐complex structure of the former. It comes as a mixed‐pullback theorem where two ‐homomorphisms are covariantly induced from path homomorphisms of graphs and the remaining two are contravariantly induced by admissible inclusions of graphs. As a main result and an application of new covariant‐induction tools, we prove such a mixed‐pullback theorem for arbitrary graphs whose all vertex‐simple loops have exits, which substantially enlarges the scope of examples coming from noncommutative topology.

To unravel the structure of fundamental examples studied in noncommutative topology, we prove that the graph C^* -algebra C^*(E) of a trimmable graph E is U(1) -equivariantly isomorphic to a pullback C^* -algebra of a subgraph C^* -algebra C^*(E'') and the C^* -algebra of functions on a circle tensored with another subgraph C^* -algebra C^*(E') . This allows us to approach the structure and K-theory of the fixed-point subalgebra C^*(E)^{U(1)} through the (typically simpler) C^* -algebras C^*(E') , C^*(E'') and C^*(E'')^{U(1)} . As examples of trimmable graphs, we consider one-loop extensions of the standard graphs encoding respectively the Cuntz algebra \mathcal{O}_2 and the Toeplitz algebra \mathcal{T} . Then we analyze equivariant pullback structures of trimmable graphs yielding the C^* -algebras of the Vaksman–Soibelman quantum sphere S^{2n+1}_q and the quantum lens space L_q^3(l;1,l) , respectively.

We study pushouts in the category of directed graphs. Our first result is that the subcategory given by strongly admissible monomorphisms is closed with respect to pushouts. Next, we determine a subcategory for which the construction of path algebras yields a contravariant functor into the category of algebras, and then we restrict this subcategory even further to ensure that now the construction of Leavitt path algebras induces a contravariant functor into the category of algebras. Finally, we prove our main theorems stating under which conditions these two contravariant functors turn pushouts of directed graphs into pullbacks of algebras.

We consider the class of directed graphs with $N\geq 1$ edges and without loops shorter than k. Using the concept of a labelled graph, we determine graphs from this class that maximize the number of all paths of length k. Then we show an R-labelled version of this result for semirings R contained in the semiring of non-negative real numbers and containing the semiring of non-negative rational numbers. We end by posing a related open problem concerning the maximal dimension of the path algebra of an acyclic graph with $N\geq1$ edges.

Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

We find a substantial class of pairs of ∗‐homomorphisms between graph C*‐algebras of the form 𝐶∗(𝐸)↪𝐶∗(𝐺)↞𝐶∗(𝐹) whose pullback C*‐algebra is an AF graph C*‐algebra. Our result can be interpreted as a recipe for determining the quantum space obtained by shrinking a quantum subspace. There are numerous examples from noncommutative topology, such as quantum complex projective spaces (including the standard Podleś quantum sphere) and quantum teardrops, that instantiate the result. Furthermore, to go beyond AF graph C*-algebras, we consider extensions of graphs over sinks and prove an analogous theorem for the thus obtained graph C*‐algebras.

We construct distinguished free generators of the $K_0$-group of the C*-algebra $C(\mathbb{CP}^n_\mathcal{T})$ of the multipullback quantum complex projective space. To this end, first we prove a quantum-tubular-neighborhood lemma to overcome the difficulty of the lack of an embedding of $\mathbb{CP}^{n-1}_\mathcal{T}$ in $\mathbb{CP}^n_\mathcal{T}$. This allows us to compute $K_0(C(\mathbb{CP}^n_\mathcal{T} ))$ using the Mayer-Vietoris six-term exact sequence in K-theory. The same lemma also helps us to prove a comparison theorem identifying the $K_0$-group of the C*-algebra $C(\mathbb{CP}^n_q)$ of the Vaksman-Soibelman quantum complex projective space with $K_0(C(\mathbb{CP}^n_\mathcal{T}))$. Since this identification is induced by the restriction-corestriction of a U(1)-equivariant \mbox{*-homomorphism} from the C*-algebra $C(S^{2n+1}_q)$ of the $(2n\!+\!1)$-dimensional Vaksman-Soibelman quantum sphere to the C*-algebra $C(S^{2n+1}_H)$ of the $(2n\!+\!1)$-dimensional Heegaard quantum sphere, we conclude that there is a basis of $K_0(C(\mathbb{CP}^n_\mathcal{T}))$ given by associated noncommutative vector bundles coming from the same representations that yield an associated-noncommutative-vector-bundle basis of the $K_0(C(\mathbb{CP}^n_q))$. Finally, using identities in K-theory afforded by Toeplitz projections in $C(\mathbb{CP}^n_\mathcal{T})$, we prove noncommutative Atiyah-Todd identities.