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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.

The quantum lens spaces form a natural and well-studied class of noncommutative spaces which has been partially classified using algebraic invariants drawing on the developed classification theory of graph $C^*$-algebras. We introduce the problem of deciding when two quantum lens spaces are equivariantly isomorphic, and solve it in certain basic cases. The results can be formulated directly in terms of the parameters defining the quantum lens spaces, and here occasionally take on a rather complicated from 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.

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., Comment: 19 pages. This version matches the version in Ergodic Theory and Dynamical Systems

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., Comment: Final version. Streamlined introduction, improved exposition, and fixed several typos. To appear in Analysis and PDEs. 37 pages

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., Comment: 17 pages

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., Comment: This is a very old paper which I am informed has attracted some current interest. Since the proceedings in which it is published are hard to come by, I have posted it here. Apart from updating the references and fixing some issues hailing from the tex-file being 25 years old, the paper is as in the published version

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.

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.

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., Comment: 1 figure, 28 pages. To appear in Compositio Mathematica

We give a complete $K$-theoretical description of when an extension of two simple graph $C^{*}$-algebras is again a graph $C^{*}$-algebra., Comment: Accepted version, to appear in Annals of K-theory

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., Comment: The results in section 4.2 (finitely generated torsion-free 2-step nilpotent groups) have been strengthened slightly. Clarified in the introduction that we only consider unitary group representations. A few misprints fixed. 39 pages

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., Comment: This article draws heavily on results and notation developed in arXiv:1602.03709, arXiv:1604.05439 and arXiv:1605.06153, and together with these papers supersedes the results of arXiv:1505.06773, which will not be published. The second version adjusts the proof of decidability in Section 14.2 to the appeared version of [BS18], corrects the statement of Corollary 3.6, and updates references

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., Comment: 25 pages. Minor changes have been made and the list of references has been updated. This is the version that will be published

We introduce filtered algebraic $K$-theory of a ring $R$ relative to a sublattice of ideals. This is done in such a way that filtered algebraic $K$-theory of a Leavitt path algebra relative to the graded ideals is parallel to the gauge invariant filtered $K$-theory for graph $C^*$-algebras. We apply this to verify the Abrams-Tomforde conjecture for a large class of finite graphs., Comment: 16 pages

We characterize when there exists a diagonal preserving $*$-isomorphism between two graph $C^*$-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of "orbit equivalence" between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal preserving $*$-isomorphism between the graph $C^*$-algebras $C^*(E)$ and $C^*(F)$.

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 Szyma\'nski, and to complete the classification of graph $C^*$-algebras associated to all simple graphs with four vertices or less., Comment: Corrected typos, corrected minor errors in statements and proofs of some results, and added Lemma 6.6

We show that the Cuntz splice induces stably isomorphic graph $C^*$-algebras., Comment: Our arguments to prove invariance of the Cuntz splice for unital graph C*-algebras in arXiv:1505.06773 applied with only minor changes in the general case. Since most of the results of that preprint have since been superseded by other forthcoming work, we do not intend to publish it, whereas this work is intended for publication. arXiv admin note: substantial text overlap with arXiv:1505.06773

We give a complete description of which unital graph C*-algebras are semiprojective, and use it to disprove two conjectures by Blackadar. To do so, we perform a detailed analysis of which projections are properly infinite in such C*-algebras.

We generalize the classification result of Restorff on Cuntz-Krieger algebras to cover all unital graph C*-algebras with real rank zero, showing that Morita equivalence in this case is determined by ordered, filtered K-theory as conjectured by three of the authors. 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 technical step, we prove that the so-called Cuntz splice leaves unital graph C*-algebras invariant up to Morita equivalence. We note that we have recently found a way to generalize the results of the present paper to cover general unital graph C*-algebras. The improved methods needed render some parts of the present paper obsolete, and hence we do not intend to publish it. Instead, we will present a complete solution (drawing heavily on many of the methods presented here) in a forthcoming paper., Comment: An inconsequential error concerning our "standard form" has been corrected

As recently pointed out by Gabe, a fundamental paper by Elliott and Kucerovsky concerning the absorption theory for C*-algebras contains an error, and as a consequence we must report that Lemma 4.5 in "Classifying C*-algebras with both finite and infinite subquotients" J. Funct. Anal. 265 (2013), no. 3, 449--468 is not true as stated. In this corrigendum, we prove an adjusted statement and explain why the error has no consequences to the main results of that paper. In particular, it is noted that all the authors' claims concerning Morita equivalence or stable isomorphism of graph C*-algebras remain correct as stated., Comment: 5 pages

Semigroup C*-algebras for right-angled Artin monoids were introduced and studied by Crisp and Laca. In the paper at hand, we are able to present the complete answer to their question of when such C*-algebras are isomorphic. The answer to this question is presented both in terms of properties of the graph defining the Artin monoids as well as in terms of classification by K-theory, and is obtained using recent results from classification of non-simple C*-algebras. Moreover, we are able to answer another natural question: Which of these semigroup C*-algebras for right-angled Artin monoids are isomorphic to graph algebras? We give a complete answer, and note the consequence that many of the C*-algebras under study are semiprojective., Comment: 32 pages, 2 figures

We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KKE(e, e'), with one specified ideal, into Hom_{\Lambda} (\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category N with the K-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K_{1}-groups of the quotients being free abelian groups. This class includes all Cuntz-Krieger algebras with exactly one non-trivial ideal. Combining our results with the results of Kirchberg, we classify automorphisms of the stabilized purely infinite Cuntz-Krieger algebras with exactly one non-trivial ideal modulo asymptotically unitary equivalence. We also get a classification result modulo approximately unitary equivalence. The results in this paper also apply to certain graph algebras., Comment: 26 pages

We consider the problem of identifying exactly which AF-algebras are isomorphic to a graph C*-algebra. We prove that any separable, unital, Type I C*-algebra with finitely many ideals is isomorphic to a graph C*-algebra. This result allows us to prove that a unital AF-algebra is isomorphic to a graph C*-algebra if and only if it is a Type I C*-algebra with finitely many ideals. We also consider nonunital AF-algebras that have a largest ideal with the property that the quotient by this ideal is the only unital quotient of the AF-algebra. We show that such an AF-algebra is isomorphic to a graph C*-algebra if and only if its unital quotient is Type I, which occurs if and only if its unital quotient is isomorphic to M_k for some natural number k. All of these results provide vast supporting evidence for the conjecture that an AF-algebra is isomorphic to a graph C*-algebra if and only if each unital quotient of the AF-algebra is Type I with finitely many ideals, and bear relevance for the intrigiung question of finding K-theoretical criteria for when an extension of two graph C*-algebras is again a graph C*-algebra., Comment: 24 pages

We show that certain extensions of classifiable C*-algebra are strongly classified by the associated six-term exact sequence in K-theory together with the positive cone of K_{0}-groups of the ideal and quotient. We apply our result to give a complete classification of graph C*-algebras with exactly one ideal., Comment: 29 pages

We describe the status quo of the classification problem of graph C*-algebras with four primitive ideals or less.

In this paper, we define an invariant, which we believe should be the substitute for total K-theory in the case when there is one distinguished ideal. Moreover, some diagrams relating the new groups to the ordinary K-groups with coefficients are constructed. These diagrams will in most cases help to determine the new groups, and will in a companion paper be used to prove a universal multi-coefficient theorem for the one distinguished ideal case for a large class of algebras.

There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C*-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of K-groups by splicing together smaller graphs whose C*-algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C*-algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C*-algebras under investigations have more than one ideal and where there are currently no relevant classification theories available., Comment: 40 pages

We provide a complete invariant for graph C*-algebras which are amplified in the sense that whenever there is an edge between two vertices, there are infinitely many. The invariant used is the standard primitive ideal space adorned with a map into {-1,0,1,2,...}, and we prove that the classification result is strong in the sense that isomorphisms at the level of the invariant always lift. We extend the classification result to cover more graphs, and give a range result for the invariant (in the vein of Effros-Handelman-Shen) which is further used to prove that extensions of graph C*-algebras associated to amplified graphs are again graph C*-algebras of amplified graphs., Comment: 15 pages, 1 figure

Let A be a C*-algebra with real rank zero which has the stable weak cancellation property. Let I be an ideal of A such that I is stable and satisfies the corona factorization property. We prove that 0->I->A->A/I->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 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., Comment: Version IV: No changes to the text. We only report that Theorem 4.9 is not correct as stated. See arXiv:1505.05951 for more details. Since Theorem 4.9 is an application to the main results of the paper, the main results of this paper are not affected by the error. Version III comments: Some typos and errors corrected. Some references added

Let $C^*(E)$ be the graph $C^*$-algebra associated to a graph E and let J be a gauge invariant ideal in $C^*(E)$. We compute the cyclic six-term exact sequence in $K$-theory of the associated extension in terms of the adjacency matrix associated to $E$. The ordered six-term exact sequence is a complete stable isomorphism invariant for several classes of graph $C^*$-algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, infinite collections of such sequences comprise complete invariants. Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of $E$.

We give a classification result for a certain class of $C^{*}$-algebras $\mathfrak{A}$ over a finite topological space $X$ in which there exists an open set $U$ of $X$ such that $U$ separates the finite and infinite subquotients of $\mathfrak{A}$. We will apply our results to $C^{*}$-algebras arising from graphs., Comment: Version III: No changes to the text. We only report that Lemma 4.5 is not correct as stated. See arXiv:1505.05951 for the corrected version of Lemma 4.5. As noted in arXiv:1505.05951, the main results of this paper are true verbatim. Version II: Improved some results in Section 3 and loosened the assumptions in Definition 4.1

We prove that a graph C*-algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by its associated six-term exact sequence in K-theory. We prove that a similar classification also holds for a graph C*-algebra with a largest proper ideal that is an AF-algebra. Our results are based on a general method developed by the first named author with Restorff and Ruiz. As a key step in the argument, we show how to produce stability for certain full hereditary subalgebras associated to such graph C*-algebras. We further prove that, except under trivial circumstances, a unique proper nontrivial ideal in a graph C*-algebra is stable., Comment: 27 pages, uses XY-pic; Version II comments: A few minor typos corrected

In this paper we show that the reducibility structure of several covers of sofic shifts is a flow invariant. In addition, we prove that for an irreducible subshift of almost finite type the left Krieger cover and the past set cover are reducible. We provide an example which shows that there are non almost finite type shifts which have reducible left Krieger covers. As an application we show that the Matsumoto algebra of an irreducible, strictly sofic shift of almost finite type is not simple., Comment: 16 pages, 5 figures

A universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory was obtained in the fundamental case of a C*-algebra with one specified ideal by Bonkat and proved there to split, unnaturally, under certain conditions. Employing certain K-theoretical information derivable from the given operator algebras in a way introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the K-theory which must be used to classify *-isomorphisms for purely infinite C*-algebras with one non-trivial ideal., Comment: 14 pages, minor typos fixed, 5 figures added

We classify extensions of certain classifiable C*-algebras using the six term exact sequence in K-theory together with the positive cone of the K_0-groups of the distinguished ideal and quotient. We then apply our results to a class of C*-algebras arising from substitutional shift spaces., Comment: 22 pages, Reordered some sections, an application involving graph algebras is added

We present the first range result for the total K-theory of C*-algebras. This invariant has been used successfully to classify certain separable, nuclear C*-algebras of real rank zero. Our results complete the classification of the so-called AD algebras of real rank zero., Comment: 24 pages

The concept of regularity in the meta-topological setting of projections in the double dual of a C*-algebra addresses the interrelations of a projection p with its closure, for instance in the form that such projections act identically, in norm, on elements of the C*-algebra. This concept has been given new actuality with the recent plan of Peligrad and Zsido to find a meaningful notion of Murray-von Neumann type equivalence among open projections. Although automatic in the commutative case, it has been known since the late sixties that regularity fails for many projections. The original investigations, however, did not answer a question such as: "Are all open and dense projections regular in A, when A is simple?" We report here that this and related questions have negative answers. In the other direction, we supply positive results on regularity of large open projections.

The soft tori constitute a continuous deformation, in a very precise sense, from the commutative C*-algebra C(T^2) to the highly non-commutative C*-algebra C*(F_2). Since both of these C*-algebras are known to have a separating family of finite dimensional representations, it is natural to ask whether that is also the case for the soft tori. We show that this is in fact the case.

The mid-seventies' works on C*-algebras of Brown-Douglas-Fillmore and Elliott both contained uniqueness and existence results in a now standard sense. These papers served as keystones for two separate theories -- KK-theory and the classification program -- which for many years parted ways with only moderate interaction. But recent years have seen a fruitful interaction which has been one of the main engines behind rapid progress in the classification program. In the present paper we take this interaction even further. We prove general existence and uniqueness results using KK-theory and a concept of quasidiagonality for representations. These results are employed to obtain new classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin. An important novel feature of these classes is that they are defined by a certain local approximation property, rather than by an inductive limit construction. Our existence and uniqueness results are in the spirit of classical Ext-theory. The main complication overcome in the paper is to control the stabilization which is necessary when one works with finite C*-algebras. In the infinite case, where programs of this type have already been successfully carried out, stabilization is unnecessary. Yet, our methods are sufficiently versatile to allow us to reprove, from a handful of basic results, the classification of purely infinite nuclear C*-algebras of Kirchberg and Phillips. Indeed, it is our hope that this can be the starting point of a unified approach to classification of nuclear C*-algebras.