Stable Isomorphism Research Articles

W-locally isomorphic torsionless modules over weakly Matlis domains

Let R be an integral domain. An R-module G is torsionless if it is isomorphic to a submodule of a finitely generated free R-module. For torsionless modules, we define w-weak isomorphism types which are new versions of weak isomorphism types with respect to the w-operation: w-local isomorphism and w-nearly isomorphism. Furthermore, we examine the relationship between w-local isomorphism, w-nearly isomorphism, and stable isomorphism for torsionless w-modules over weakly Matlis domains.

Decidability of flow equivalence and isomorphism problems for graph C*-algebras and quiver representations

We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z (the integers). Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).

Stable Isomorphisms of Operator Algebras

Abstract Let ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ be operator algebras with $c_{0}$-isomorphic diagonals and let ${{\mathcal{K}}}$ denote the compact operators. We show that if ${{\mathcal{A}}}\otimes{{\mathcal{K}}}$ and ${{\mathcal{B}}}\otimes{{\mathcal{K}}}$ are isometrically isomorphic, then ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are isometrically isomorphic. If the algebras ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ satisfy an extra analyticity condition a similar result holds with ${{\mathcal{K}}}$ being replaced by any operator algebra containing the compact operators. For nonselfadjoint graph algebras this implies that the graph is a complete invariant for various types of isomorphisms, including stable isomorphisms, thus strengthening a recent result of Dor-On, Eilers, and Geffen. Similar results are proven for algebras whose diagonals satisfy cancellation and have $K_{0}$-groups isomorphic to ${{\mathbb{Z}}}$. This has implications in the study of stable isomorphisms between various semicrossed products.

Morita equivalence of w*-rigged modules

The w*-rigged modules over dual operator algebras were introduced by Blecher and Kashyap as a generalization of W*-modules. In this paper, we introduce two new types of Morita equivalence between right w*-rigged modules over unital dual operator algebras and we examine whether these notions imply stable isomorphism between the corresponding modules. Furthermore, we investigate them in detail for the class of right w*-rigged modules over nest algebras, a class which was characterized by G.K. Eleftherakis.

Rigidity results for automorphisms of Hardy--Toeplitz C∗-algebras

We prove a number of results on the automorphisms and isomorphisms between Hardy--Toeplitz algebras T(D) associated to bounded symmetric domains D: that the stable isomorphism class of T(D) determines D (even when it is reducible), that for reducible domains D=D1×⋯×Ds the automorphisms of the Shilov boundary ˇS(D) induced by those of T(D) permute the Shilov boundaries ˇS(Di), and that by contrast to arbitrary solvable algebras, automorphisms of T(D) that are trivial on their character spaces ˇS(D) are trivial on the entire spectrum ˆT(D).

Stabilising uniform property $\Gamma $

We introduce stabilised property Gamma, a C*-algebraic variant of property Gamma which is invariant under stable isomorphism. We then show that simple separable nuclear C*-algebras with stabilised property Gamma and $\mathrm{Cu}(A) \cong \mathrm{Cu}(A \otimes \mathcal{Z})$ absorb the Jiang-Su algebra $\mathcal{Z}$ tensorially.

Locally isomorphic torsionless modules over domains of finite character

In a 2002 paper, P. Goeters and B. Olberding compare local, near, and stable isomorphisms of torsionless modules over [Formula: see text]-local domains. In this paper, we compare these weaker forms of isomorphisms of torsionless modules over domains of finite character.

Finite formal model of toric singularities

We study the formal neighborhoods at rational non-degenerate arcs of the arc scheme associated with a toric variety. The first main result of this article shows that these formal neighborhoods are generically constant on each Nash component of the variety. Furthermore, using our previous work, we attach to every such formal neighborhood, and in fact to every toric valuation, a minimal formal model (in the class of stable isomorphisms) which can be interpreted as a measure of the singularities of the base-variety. As a second main statement, for a large class of toric valuations, we compute the dimension and the embedding dimension of such minimal formal models, and we relate the latter to the Mather discrepancy. The class includes the strongly essential valuations, that is to say those the center of which is a divisor in the exceptional locus of every resolution of singularities of the variety. We also obtain a similar result for monomial curves.

State splitting, strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras

ABSTRACTWe prove that if two nonnegative matrices are strong shift equivalent, the associated stable Cuntz–Krieger algebras with generalized gauge actions are conjugate. The proof is done by a purely functional analytic method and based on constructing imprimitivity bimodule from bipartite directed graphs through strong shift equivalent matrices, so that we may clarify K-theoretic behaviour of the stable isomorphism between the associated stable Cuntz–Krieger algebras. We also examine our machinery for the matrices obtained by state splitting graphs, so that topological conjugacy of the topological Markov shifts is described in terms of some equivalence relation of the Cuntz–Krieger algebras with canonical masas and the gauge actions without stabilization.

Geometric Classification of Graph C*-algebras over Finite Graphs

AbstractWe 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 uncountably many ideals.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 isomorphicK-theories. This proves in turn that under this condition, the graph C*-algebras are in fact classifiable byK-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 quantumlens spaces defined by Hong and Szymański, and to complete the classification of graph C*-algebras associated with all simple graphs with four vertices or less.

Isomorphism of inclusion compounds built with composite host lattices of thiadiazole and V-shaped molecules with the existence of tetrabutylammonium guest template

ABSTRACTBy using 4,4′-dihydroxydiphenylsulfone (BPS) and 4,4′-dihydroxydiphenylsulfide (BTS) react with amidinethiourea and tetrabutylammonium respectively, two isomorphs of inclusion compounds of (n-C4H9)4N+·C12H9O4S−·C2H4N4S (1) and (n-C4H9)4N+·C12H9O2S−·C2H4N4S (2) were prepared and characterized by Single crystal X-ray diffraction. Although two varied types of V-shaped molecules were selected to react in the actual experimental process, two isomorphs with similar packing modes were finally obtained. Interestingly, amidinothiourea molecule occurred in situ oxidation reaction to generate a new compound of 3,5-dimido-1,2,4,-thiadiazole during the experiments, and these two inclusion compounds here are the pioneer examples of thiadiazole and tetraalkylammonium and can be regarded as two isomorphs due to their similar crystallization modes. In these two isomorphs, thiadiazole molecules derived from amidinothiourea link with various V-type molecules to develop the similar dumbbell-type hydrogen-bonded ribbons by the alike hydrogen bonding modes, then the ribbons interpenetrate with tetrabutylammonium to result in two stable isomorphism structures.

Cuntz Splice invariance for purely infinite graph algebras

We show that the Cuntz Splice preserves the stable isomorphism class of a purely infinite graph $\mathrm{C}^*$-algebra with finitely many ideals.

Some remarks on L-equivalence of algebraic varieties

In this short note we study the questions of (non-)L-equivalence of algebraic varieties, in particular, for abelian varieties and K3 surfaces. We disprove the original version of a conjecture of Huybrechts (Int J Math 16(1):13–36, 2005, Conjecture 0.3) stating that isogenous K3 surfaces are L-equivalent. Moreover, we give examples of derived equivalent twisted K3 surfaces, such that the underlying K3 surfaces are not L-equivalent. We also give examples showing that D-equivalent abelian varieties can be non-L-equivalent (the same examples were obtained independently in Ito et al. Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties. arXiv:1612.08497 ). This disproves the original version of a conjecture of Kuznetsov and Shinder (Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics. Sel Math New Ser. arXiv:1612.07193 , Conjecture 1.6). We deduce the statements on (non-)L-equivalence from the very general results on the Grothendieck group of an additive category, whose morphisms are finitely generated abelian groups. In particular, we show that in such a category each stable isomorphism class of objects contains only finitely many isomorphism classes. We also show that a stable isomorphism between two objects X and Y with $${\text {End}}(X)=\mathbb {Z}$$ implies that X and Y are isomorphic.

Morita equivalence of C*-correspondences passes to the related operator algebras

We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz–Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong Δ-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a σ-TRO. As an application we show that strong Morita equivalence coincides with strong Δ-equivalence for tensor algebras of aperiodic C*-correspondences.

C*-Envelopes of Tensor Algebras Arising from Stochastic Matrices

In this paper we study the C*-envelope of the (non-self-adjoint) tensor algebra associated via subproduct systems to a finite irreducible stochastic matrix P. Firstly, we identify the boundary representations of the tensor algebra inside the Toeplitz algebra, also known as its non-commutative Choquet boundary. As an application, we provide examples of C*-envelopes that are not *-isomorphic to either the Toeplitz algebra or the Cuntz–Pimsner algebra. This characterization required a new proof for the fact that the Cuntz–Pimsner algebra associated to P is isomorphic to $$C({\mathbb {T}}, M_d({\mathbb {C}}))$$ , filling a gap in a previous paper. We then proceed to classify the C*-envelopes of tensor algebras of stochastic matrices up to *-isomorphism and stable isomorphism, in terms of the underlying matrices. This is accomplished by determining the K-theory of these C*-algebras and by combining this information with results due to Paschke and Salinas in extension theory. This classification is applied to provide a clearer picture of the various C*-envelopes that can land between the Toeplitz and the Cuntz–Pimsner algebras.

Strong classification of purely infinite Cuntz-Krieger algebras

In 2006, Restorff completed the classification of all Cuntz-Krieger algebras with finitely many ideals (i.e., those that are purely infinite) up to stable isomorphism. He left open the questions concerning strong classification up to stable isomorphism and unital classification. In this paper, we address both questions. We show that any isomorphism between the reduced filtered K K -theory of two Cuntz-Krieger algebras with finitely many ideals lifts to a ∗ ^* -isomorphism between the stabilized Cuntz-Krieger algebras. As a result, we also obtain strong unital classification.

Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph 𝐶*-algebras and Leavitt path algebras

We prove that ample groupoids with σ \sigma -compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui’s notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C ∗ C^* -algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras L Z ( E 2 ) L_{\mathbb {Z}}(E_2) and L Z ( E 2 − ) L_{\mathbb {Z}}(E_{2-}) are not stably ∗ ^* -isomorphic.

Unique prime factorization and bicentralizer problem for a class of type III factors

We show that whenever m≥1 and M1,…,Mm are nonamenable factors in a large class of von Neumann algebras that we call C(AO) and which contains all free Araki–Woods factors, the tensor product factor M1⊗‾⋯⊗‾Mm retains the integer m and each factor Mi up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [33,25] and moreover provides new UPF results in the case when M1,…,Mm are free Araki–Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type III1 factors in the class C(AO).

Strong Morita equivalence of operator spaces

We introduce and examine the notions of strong Δ-equivalence and strong TRO equivalence for operator spaces. We show that they behave in an analogous way to how strong Morita equivalence does for the category of C*-algebras. In particular, we prove that strong Δ-equivalence coincides with stable isomorphism under the expected countability hypothesis, and that strongly TRO equivalent operator spaces admit a correspondence between particular representations. Furthermore we show that strongly Δ-equivalent operator spaces have stably isomorphic second duals and strongly Δ-equivalent TRO envelopes. In the case of unital operator spaces, strong Δ-equivalence implies stable isomorphism of the C*-envelopes.

Galois structure ofS-units

Let K/k be a finite Galois extension of number fields with Galois group G, S a large set of primes of K, and E the G-module of S-units of K. Previous work has determined the data which is necessary to determine the stable isomorphism class of E. This paper explains how to build a G-module M from this data which has the property that M is stably isomorphic to the direct sum of E and some free G-module F.