Order Isomorphism Research Articles

The operator system of Toeplitz matrices

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of n × n n\times n Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than n n . The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the n × n n\times n complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the n × n n\times n Toeplitz matrices into the algebra of all n × n n\times n complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of n × n n\times n complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix ξ n \xi _n generates an extremal ray in the cone of all continuous n × n n\times n Toeplitz-matrix valued functions f f on the unit circle S 1 S^1 whose Fourier coefficients f ^ ( k ) \hat f(k) vanish for | k | ≥ n |k|\geq n . Lastly, it is noted that all positive Toeplitz matrices over nuclear C ∗ ^* -algebras are approximately separable.

The range of ultrametrics, compactness, and separability

We describe the order type of range sets of compact ultrametrics and show that an ultrametrizable infinite topological space (X,τ) is compact iff the range sets are order isomorphic for any two ultrametrics compatible with the topology τ. It is also shown that an ultrametrizable topology is separable iff every ultrametric, compatible with this topology, has at most countable range set.

Radicals of principal ideals and the class group of a Dedekind domain

For a Dedekind domain $D$, let $\mathcal{P}(D)$ be the set of ideals of $D$ that are radical of a principal ideal. We show that, if $D,D'$ are Dedekind domains and there is an order isomorphism between $\mathcal{P}(D)$ and $\mathcal{P}(D')$, then the rank of the class groups of $D$ and $D'$ is the same.

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, we establish that they leave the graph algebras invariant, and we prove that after this augmentation, the list of moves becomes complete in the sense described above. Along the way, we prove that every (reduced, filtered) K-theory order isomorphism can be lifted to an isomorphism between the stabilized C∗-algebras—and, as a consequence, that every such order 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 and ∗-isomorphism among 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 we verify, in the unital case, the Abrams–Tomforde conjectures.

Analytic order-isomorphisms of countable dense subsets of the unit circle

AbstractFor functions in $C^k(\mathbb {R})$ which commute with a translation, we prove a theorem on approximation by entire functions which commute with the same translation, with a requirement that the values of the entire function and its derivatives on a specified countable set belong to specified dense sets. Using this theorem, we show that if A and B are countable dense subsets of the unit circle $T\subseteq \mathbb {C}$ with $1\notin A$ , $1\notin B$ , then there is an analytic function $h\colon \mathbb {C}\setminus \{0\}\to \mathbb {C}$ that restricts to an order isomorphism of the arc $T\setminus \{1\}$ onto itself and satisfies $h(A)=B$ and $h'(z)\not =0$ when $z\in T$ . This answers a question of P. M. Gauthier.

The projective geometry over partially ordered skew fields, II

In this paper > the investigation of properties for partially ordered linear spaces over partially ordered skew fields is prolonged. This investigation was started in part I >. Derivative lattices associated partially ordered linear spaces over partially ordered skew fields are examined. More exactly, properties of the convex projective geometry $${\cal L}$$ of a partially ordered linear space $${}_FV$$ over a partially ordered skew field $$F$$ are considered. The convexity of linear subspaces has meaning the Abelian convexity ($$ab$$-convexity), which is based on the definition of a convex subgroup for a partially ordered group. Second and third theorems of linear spaces order isomorphisms for interpolation linear spaces over partially ordered skew fields are proved. Some theorems are proved for principal linear subspaces of interpolation linear spaces over directed skew fields. The principal linear subspace $$I_a$$ of a partially ordered linear space $${}_FV$$ over a partially ordered skew field $$F$$ is the smallest $$ab$$-convex directed linear subspace of linear space $${}_FV$$ which contains the positive element $$a\in V$$. The analog for the third theorem of linear spaces order isomorphisms for principal linear subspaces is demonstrated in interpolation linear spaces over directed skew fields

Mining frequent approximate patterns in large networks

AbstractFrequent pattern mining (FPM) algorithms are often based on graph isomorphism in order to identify common pattern occurrences. Recent research works, however, have focused on cases in which patterns can differ from their occurrences. Such cases have great potential for the analysis of noisy network data. Most existing FPM algorithms consider differences in edges and their labels, but none of them so far has considered the structural differences of vertices and their labels. Discerning how to identify cases that differ from the initial pattern by any number of vertices, edges, or labels has become the main challenge of recent research works. As a solution, we suggest a novel FMP algorithm named mining frequent approximate patterns (MFAPs) with two central new characteristics. First, we begin by using the inexact matching technique, which allows for structural differences in edge, vertices, and labels. Second, we follow the approximate matching with a focus on mining patterns within the directed graph, as opposed to the more commonly explored case of patterns being mined from the undirected graph. Our results illustrate the effectiveness of this new MFAP algorithm in identifying patterns within an optimized time.

On a variant of Tingley's problem for some function spaces

Let (Ω,A,μ) and (Γ,B,ν) be two arbitrary measure spaces, and p∈[1,∞]. SetS(Lp(μ))+:={f∈Lp(μ):‖f‖p=1;f≥0μ-a.e.}, that is, the positive part of the unit sphere of Lp(μ). We show that every surjective isometry Φ:S(Lp(μ))+→S(Lp(ν))+ can be extended (necessarily uniquely) to an isometric order isomorphism from Lp(μ) onto Lp(ν). A Lamperti form, i.e., a weighted composition like form, of Φ is provided, when (Γ,B,ν) is localizable (in particular, when it is σ-finite). On the other hand, we show that for compact Hausdorff spaces X and Y, if Φ is a surjective isometry from the positive part of the unit sphere of C(X) to that of C(Y), then there is a homeomorphism τ:Y→X satisfying Φ(f)(y)=f(τ(y)) for f∈S(C(X))+ and y∈Y.

Complete order equivalence of spin unitaries

This paper is a study of linear spaces of matrices and linear maps on matrix algebras that arise from spin systems, or spin unitaries, which are finite sets S of selfadjoint unitary matrices such that any two unitaries in S anticommute. We are especially interested in linear isomorphisms between these linear spaces of matrices such that the matricial order within these spaces is preserved; such isomorphisms are called complete order isomorphisms, which might be viewed as weaker notion of unitary similarity. The main result of this paper shows that all m-tuples of anticommuting selfadjoint unitary matrices are equivalent in this sense, meaning that there exists a unital complete order isomorphism between the unital linear subspaces that these tuples generate. We also show that the C⁎-envelope of any operator system generated by a spin system of cardinality 2k or 2k+1 is the simple matrix algebra M2k(C). As an application of the main result, we show that the free spectrahedra determined by spin unitaries depend only upon the number of the unitaries, not upon the particular choice of unitaries, and we give a new, direct proof of the fact [13] that the spin ball Bmspin and max ball Bmmax coincide as matrix convex sets in the cases m=1,2. We also derive analogous results for countable spin systems and their C⁎-envelopes.

Order Isomorphisms on Order Intervals of Atomic JBW-Algebras

In this paper a full description of order isomorphisms between effect algebras of atomic JBW-algebras is given. We will derive a closed formula for the order isomorphisms on the effect algebra of type I factors by proving that the invertible part of the effect algebra of a type I factor is left invariant. This yields an order isomorphism on the whole cone, for which a characterisation exists. Furthermore, we will show that the obtained formula for the order isomorphism on the invertible part can be extended to the whole effect algebra again. As atomic JBW-algebras are direct sums of type I factors and order isomorphisms factor through the direct sum decomposition, this yields the desired description.

Order isomorphisms between cones of JB-algebras

We completely describe the order isomorphisms between the cones of atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an algebraic direct summand of the so-called engaged and disengaged part. On the cone of the engaged part every order i

On two new classes of stabilizers in residuated lattices

In this paper, we introduce some stabilizers and study related properties of them in residuated lattices. Then, we investigate the image and inverse image of a right and left stabilizer of a nonempty subset under a homomorphism. Besides, we discuss the relations between stabilizers and several special filters (ideals) in residuated lattices. Moreover, we also characterize some special classes of residuated lattices, for example, Heyting algebras and linearly ordered Heyting algebras, in terms of these stabilizers. Finally, we discuss the relations between these stabilizers and get that the right implicative stabilizers and right multiplicative stabilizers are order isomorphic.

Jordan Invariants of Von Neumann Algebras Given by Abelian Subalgebras and Choquet Order on State Spaces

New complete invariants for Jordan parts of von Neumann algebras are presented. We shall prove that the poset of all finite dimensional abelian von Neumann subalgebras ordered by set theoretic inclusion is a complete Jordan invariant for von Neumann algebras. On the other hand, we exhibit an example showing that not any order isomorphism on this structure is derived from a Jordan isomorphism. We apply our results to the Choquet order of orthogonal measures on state spaces of von Neumann algebras. Among others we show that the poset of decompositions of a fixed faithful normal state on a von Neumann algebra endowed with the Choquet order is a complete Jordan invariant for σ-finite von Neumann algebras.

Universal Entire Functions That Define Order Isomorphisms of Countable Real Sets

AbstractIn 1895, Cantor showed that between every two countable dense real sets, there is an order isomorphism. In fact, there is always such an order isomorphism that is the restriction of a universal entire function.

The monoid of order isomorphisms between principal filters of $${\varvec{\mathbb {N}}}^n$$Nn

Let n be any positive integer and be the semigroup of all order isomorphisms between principal filters of the nth power of the set of positive integers $${\mathbb {N}}$$ with the product order. We study algebraic properties of the semigroup . In particular, we show that is a bisimple, E-unitary, F-inverse semigroup, describe Green’s relations on and its maximal subgroups. We show that the semigroup is isomorphic to the semidirect product of the direct nth power of the bicyclic monoid by the group of permutation . Also we prove that every non-identity congruence $${\mathfrak {C}}$$ on the semigroup is a group and describe the least group congruence on . We show that every Hausdorff shift-continuous topology on is discrete and discuss embedding of the semigroup into compact-like topological semigroups.

Order Isomorphisms of Operator Intervals in von Neumann Algebras

We give a complete description of order isomorphisms between operator intervals in general von Neumann algebras. For the description, we use Jordan $^*$-isomorphisms and locally measurable operators. Our results generalize several works by L. Moln\'ar and P. \v{S}emrl on type I factors.

Three characterisations of the sequential product

It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product a◦b=aba on an operator algebra. We give three additional properties, each of which characterises the standard sequential product on either a von Neumann algebra or a Euclidean Jordan algebra. These properties are (1) invariance under the application of unital order isomorphisms, (2) symmetry of the sequential product with respect to a certain inner product, and (3) preservation of invertibility of the effects. To give these characterisations, we first have to study convex σ-sequential effect algebras. We show that these objects correspond to unit intervals of spectral order unit spaces with a homogeneous positive cone.

Stabilizers in MTL-algebras

In the paper, we introduce some stabilizers and investigate related properties of them in MTL-algebras.Then, we also characterize some special classes of MTL-algebras, for example, IMTL-algebras, integral MTL-algebras, Godel algebras and MV-algebras, in terms of these stabilizers. Moreover, we discuss the relation between stabilizers and several special filters (ideals) in MTL-algebras. Finally, we discuss the relation between these stabilizers and prove that the right implicative stabilizer and right multiplicative stabilizer are order isomorphic. This results also give answers to some open problems, which were proposed by Motamed and Torkzadeh in [Soft Comput, {\bf 21} (2017) 686-693].

Extremal Unital Completely Positive Maps and Their Symmetries

We consider the set $$P_1({\mathcal A},{\mathcal M})$$ (respectively $$CP_1({\mathcal A},{\mathcal M})$$ of unital positive (completely) maps from a $$C^*$$ algebra $${\mathcal A}$$ to a von-Neumann sub-algebra $${\mathcal M}$$ of $${\mathcal B}({\mathcal H})$$ , the algebra of bounded linear operators on a Hilbert space $${\mathcal H}$$ . We study the extreme points of the convex set $$P_1({\mathcal A},{\mathcal M})$$ ( $$CP_1({\mathcal A},{\mathcal M})$$ ) via their canonical lifting to the convex set of (unital) positive (completely) normal maps from $$\hat{{\mathcal A}}$$ to $${\mathcal M}$$ , where $${\mathcal A}^{**}$$ is the universal enveloping von-Neumann algebra over $${\mathcal A}$$ . If $${\mathcal A}={\mathcal M}$$ then a (completely) positive map $$\tau $$ admits a unique decomposition into a sum of a normal and a singular (completely) positive maps. Furthermore, if $${\mathcal M}$$ is a factor then a unital complete positive map is a unique convex combination of unital normal and singular completely positive maps. We also used a duality argument to find a criteria for an element in the convex set of unital completely positive maps with a given faithful normal invariant state on $${\mathcal M}$$ to be extremal. In our investigation, gauge symmetry in the minimal Stinespring representation of a completely positive map and Kadison theorem on order isomorphism played an important role.

A new proof of Kirchberg's 𝒪2-stable classification

Abstract I present a new proof of Kirchberg’s 𝒪 2 \mathcal{O}_{2} -stable classification theorem: two separable, nuclear, stable/unital, 𝒪 2 \mathcal{O}_{2} -stable C ∗ C^{\ast} -algebras are isomorphic if and only if their ideal lattices are order isomorphic, or equivalently, their primitive ideal spaces are homeomorphic. Many intermediate results do not depend on pure infiniteness of any sort.