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Søren Eilers

Publications and source records attributed to Søren Eilers.

At least 19 recordsLinked to original sources
arXiv2025

Chromatic numbers for contact graphs of congruent cuboids

We initiate the study of chromatic numbers for contact graphs of configurations of integer-sized cuboids in three dimensions, all of which are mutually congruent. Disallowing rotations, we show a global upper bound of 8 for the chromatic numbers, which implies that there is a global upper bound of 48 when the cuboids may be rotated freely. Specializing further to cuboids that are required to have a side length of one we obtain more precise upper bounds. Such upper bounds are compared to examples of configurations having relatively large chromatic numbers, leading to a complete determination of some of these chromatic numbers, but in general, the gaps between our upper and lower bounds are rather wide. In particular, we know of no such configuration of any size leading to a chromatic number above 6.

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arXiv2025

Diagonal-preserving Isomorphisms of Algebras from Infinite Graphs

We establish logical equivalence between statements involving * the Cuntz C*-algebra $\mathcal O_\infty$ with its canonical diagonal; * graph C*-algebras with their canonical diagonals; * Leavitt path algebras over general fields with their canonical diagonals; * Leavitt path algebras over $\mathbb Z$; * topological full groups; * groupoids; and * the automorphism $x\mapsto -x$ on certain $K_0$- and homology groups equal to $\mathbb Z$ Deciding whether these equivalent statements are true or false is of importance in studies of geometric classification of diagonal-preserving isomorphism between graph C*-algebras and Leavitt path algebras, mirroring a similar hindrance studied by Cuntz more than 40 years ago.

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arXiv2025

Refined moves for structure-preserving isomorphism of graph C*-algebras

We formalize eight different notions of isomorphism among (unital) graph C*-algebras, and initiate the study of which of these notions may be described geometrically as generated by moves. We propose a list of seven types of moves that we conjecture has the property that the collection of moves respecting one of six notions of isomorphism indeed generate that notion, in the sense that two graphs are equivalent in that sense if and only if one may transform one into another using only these kinds of moves. We carefully establish invariance properties of each move on our list, and prove a collection of generation results supporting our conjecture with an emphasis on the gauge simple case. In two of the six cases, we may prove the conjecture in full generality, and in two we can show it for all graphs defining gauge simple C*-algebras. In the two remaining cases we can show the conjecture for all graphs defining gauge simple C*-algebras provided that they are either finite or have at most one vertex allowing a path back to itself.

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arXiv2025

Equivariant isomorphism of Quantum Lens Spaces of low dimension

The quantum lens spaces form a natural and well-studied class of noncommutative spaces which can be subjected to classification using algebraic invariants by drawing on the fully developed classification theory of unital graph $C^*$-algebras. We introduce the problem of deciding when two quantum lens spaces are equivariantly isomorphic, and solve it in certain basic cases. As opposed to classification up to isomorphism, we can not appeal to a complete general classification theory in the equivariant case, but by combining existing partial results with an ad hoc analysis we can solve the case with dimension 3 completely, and the case with dimension 5 in the prime case. Our results can be formulated directly in terms of the parameters defining the quantum lens spaces, and here occasionally take on a rather complicated form which convinces us that there is a deep underlying explanation for our findings. We complement the fully established partial results with computer experiments that may indicate the way forward.

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arXiv2024

Subhomogeneity in the classification of real rank zero C*-algebras

In this paper, we construct a class of ASH algebras of real rank zero and stable rank one which is not K-pure. Then we show the following: (i) There exists a real rank zero inductive limit of 1-dimensional noncommutative CW complexes which is not an A$\mathcal{HD}$ algebra, when $K_1$ is torsion free or has bounded torsion. (ii) Total K-theory is not a complete invariant for ASH algebras of real rank zero. (iii) There are obstructions both in the total K-theory of ideals and quotients in the classification of $C^*$-algebras of real rank zero and stable rank one.

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arXiv2023

Shift equivalences through the lens of Cuntz-Krieger algebras

Motivated by Williams' problem of measuring novel differences between shift equivalence (SE) and strong shift equivalence (SSE), we introduce three equivalence relations that provide new ways to obstruct SSE while merely assuming SE. Our shift equivalence relations arise from studying graph C*-algebras, where a variety of intermediary equivalence relations naturally arise. As a consequence we realize a goal sought after by Muhly, Pask and Tomforde, measure a delicate difference between SSE and SE in terms of Pimsner dilations for C*-correspondences of adjacency matrices, and use this distinction to refute a proof from a previous paper.

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arXiv2022

Conjugacy of local homeomorphisms via groupoids and C*-algebras

We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterise topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalises recent work of Matsumoto and of the second- and third-named authors.

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arXiv2021

C*-stability of discrete groups

A group may be considered $C^*$-stable if almost representations of the group in a $C^*$-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are $C^*$-stable or only stable with respect to some subclass of $C^*$-algebras, e.g. finite dimensional $C^*$-algebras. We provide criteria and invariants for stability of groups and this allows us to completely determine stability/non-stability of crystallographic groups, finitely generated torsion-free step-2 nilpotent groups, surface groups, virtually free groups and certain Baumslag-Solitar groups.

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arXiv2020

Amplified graph C*-algebras II: reconstruction

Let $E$ be a countable directed graph that is amplified in the sense that whenever there is an edge from $v$ to $w$, there are infinitely many edges from $v$ to $w$. We show that $E$ can be recovered from $C^*(E)$ together with its canonical gauge-action, and also from $L_K(E)$ together with its canonical grading.

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arXiv2020

Classification of irreversible and reversible Pimsner operator algebras

Since their inception in the 30's by von Neumann, operator algebras have been used in shedding light in many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two was sought since their emergence in the late 60's. We connect these seemingly separate type 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.

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arXiv2020

Künneth Splittings and Classification of C*-Algebras with Finitely Many Ideals

The class of AD algebras of real rank zero is classified by an exact sequence of K-groups with coefficients, equipped with certain order structures. Such a sequence is always split, and one may ask why, then, the middle group is relevant for classification. The answer is that the splitting map can not always be chosen to respect the order structures involved. This may be rephrased in terms of the ideals of the C*-algebras in question. We prove that when the C*-algebra has only finitely many ideals, a splitting map respecting these always exists. Hence AD algebras of real rank zero with finitely many ideals are classified by (classical) ordered K-theory. We also indicate how the methods generalize to the full class of ASH algebras with slow dimension growth and real rank zero.

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arXiv2019

Geometric classification of isomorphism of unital graph C*-algebras

We geometrically describe the relation induced on a set of graphs by isomorphism of their associated graph C*-algebras as the smallest equivalence relation generated by five types of moves. The graphs studied have finitely many vertices and finitely or countably infinitely many edges, corresponding to unital and separable C*-algebras.

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arXiv2019

Flow Equivalence of G-SFTs

In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G$=\mathbb Z_2$, we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata.

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arXiv2019

The complete classification of unital graph $C^*$-algebras: Geometric and strong

We provide a complete classification of the class of unital graph $C^*$-algebras - prominently containing the full family of Cuntz-Krieger algebras - showing that Morita equivalence in this case is determined by ordered, filtered $K$-theory. The classification result is geometric in the sense that it establishes that any Morita equivalence between $C^*(E)$ and $C^*(F)$ in this class can be realized by a sequence of moves leading from $E$ to $F$, in a way resembling the role of Reidemeister moves on knots. As a key ingredient, we introduce a new class of such moves, establish that they leave the graph algebras invariant, and prove that after this augmentation, the list of moves becomes complete in the sense described above. Along the way, we prove that every ordered, reduced filtered $K$-theory isomorphism can be lifted to an isomorphism between the stabilized $C^*$-algebras - and, as a consequence, that every ordered, reduced filtered $K$-theory isomorphism preserving the class of the unit comes from a $*$-isomorphism between the unital graph $C^*$-algebras themselves. It follows that the question of Morita equivalence amongst unital graph $C^*$-algebras is a decidable one. As immediate examples of applications of our results we revisit the classification problem for quantum lens spaces and verify, in the unital case, the Abrams-Tomforde conjectures.

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arXiv2018

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph $C^*$-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz-Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.

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arXiv2017

Flow equivalence and isotopy for subshifts

We study basic properties of flow equivalence on one-dimensional compact metric spaces with a particular emphasis on isotopy in the group of (self-) flow equivalences on such a space. In particular, we show that an orbit-preserving such map is not always an isotopy, but that this always is the case for suspension flows of irreducible shifts of finite type. We also provide a version of the fundamental discretization result of Parry and Sullivan which does not require that the flow maps are either injective or surjective. Our work is motivated by applications in the classification theory of sofic shift spaces, but has been formulated to supply a solid and accessible foundation for other purposes.

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arXiv2016

Geometric classification of graph $C^*$-algebras over finite graphs

We address the classification problem for graph $C^*$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph $C^*$-algebras may come with uncountable ideal structures. We find that in this generality, stable isomorphism of graph $C^*$-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the $C^*$-algebras having isomorphic $K$-theories. This proves in turn that under this condition, the graph $C^*$-algebras are in fact classifiable by $K$-theory, providing in particular complete classification when the $C^*$-algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs. Our results are applied to discuss the classification problem for the quantum lens spaces defined by Hong and Szymański, and to complete the classification of graph $C^*$-algebras associated to all simple graphs with four vertices or less.

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