Almost commutative models provide a framework for Connes' work on the standard model of particle physics. These models are constructed as products of a the canonical spectral triple of a compact connected spin manifold with a finite dimensional spectral triple. Motivated by the fundamental question of the dependence of the spectra of Dirac operators under change of metrics, we prove the continuity of the spectra of Dirac operators for almost commutative models as functions of the underlying Riemannian metric. We allow both the Riemannian metric (in the $C^1$ topology) and the Dirac operator of the finite-dimensional factor to vary simultaneously. Since the physics of the system is fundamentally encoded in this spectrum, this result is a form of stability result regarding the geometry, or physical, content of these models. This result is based upon a novel approach to prove continuity of spectra of Dirac operators using the spectral propinquity. Notably, this method provides a new, different proof of the classical results as well. To illustrate the versatility of our new method, we also apply our results to completely non-commutative family of examples, including quantum tori and quantum solenoids.
We introduce a hypertopology, induced by an inframetric up to full quantum isometry, on the class of pointed proper quantum metric spaces, which are separable, possibly non-unital, C*-algebras endowed with an analogue of the Lipschitz seminorm, with a distinguished state, and with a particular type of approximate units. Our hypertopology provides an analogue of the Gromov-Hausdorff distance on proper metric spaces, and in fact, convergence in the latter implies convergence in the former. Moreover, when restricted to the class of quantum compact metric spaces, our new topology is compatible with the topology of the Gromov-Hausdorff propinquity. We include new examples of noncompact, noncommutative pointed proper quantum metric spaces which are limits, for our new topology, of finite dimensional quantum compact metric spaces. This article thus provides a first answer to the challenging question of how to extend noncommutative metric geometry to the locally compact quantum space realm.
We prove that a polynomial path of Riemannian metrics on a closed spin manifold induces a continuous field in the spectral propinquity of metric spectral triples.
If two compact quantum metric spaces are close in the metric sense, then how similar are they, as noncommutative spaces? In the classical realm of Riemannian geometry, informally, if two manifolds are close in the Gromov-Hausdorff distance, and belong to a class of manifolds with bounded curvature and diameter, then the spectra of their Laplacian or Dirac operators are also close under many scenari. Of particular interest is the case where a sequence of manifolds converge for the Gromov-Hausdorff distance to a manifold of lower dimension, and the question of the continuity, in some sense, of the spectra of geometrically relevant operators. In this paper, we initiate the study of the continuity of spectra and other properties of metric spectral triples under collapse in the noncommutative realm. As a first step in this study, we work with collapse for the spectral propinquity, an analogue of the Gromov-Hausdorff distance for spectral triples introduced by the second author, i.e. a form of metric for differential structures. Inspired by results from collapse in Riemannian geometry, we begin with the study of spectral triples which decompose, in some sense, in a vertical and a horizontal direction, and we collapse these spectral triples along the vertical direction. We obtain convergence results, and by the work of the second author, we conclude continuity results for the spectra of the Dirac operators of these spectral triples. Examples include collapse of product of spectral triples with one Abelian factor, $U(1)$ principal bundles over Riemannian spin manifolds, and noncommutative principal bundles, including C*-crossed-products and other noncommutative bundles.
We address the natural question: as noncommutative solenoids are inductive limits of quantum tori, do the standard spectral triples on quantum tori converge to some spectral triple on noncommutative solenoid for the spectral propinquity? We answer this question by showing that, using appropriate bounded perturbation of the spectral triples on quantum tori, such a spectral triple on noncommutative solenoid can be constructed.
Given a compact quantum metric space (A, L), we prove that the domain of L coincides with A if and only if A is finite dimensional. We then show how one can explicitly build many quantum metrics with distinct domains on infinite-dimensional AF algebras. In the last section, we provide a strategy for calculating the distance between certain states in these quantum metrics, which allow us to calculate the distance between pure states in these quantum metrics on the quantized interval and on the Cantor space.
In this paper we study the groups of isometries and the set of bi-Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latremoliere. In particular we prove that these groups and sets are compact in the automorphism group of the spectral triple C*-algebra with respect to the Monge-Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov-Hausdorff propinquity -- a noncommutative analogue of the Gromov-Hausdorff distance -- when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.
We construct a new version of the dual Gromov--Hausdorff propinquity that is sensitive to the strongly Leibniz property. In particular, this new distance is complete on the class of strongly Leibniz quantum compact metric spaces. Then, given an inductive limit of C*-algebras for which each C*-algebra of the inductive limit is equipped with a strongly Leibniz L-seminorm, we provide sufficient conditions for placing a strongly Leibniz L-seminorm on an inductive limit such that the inductive sequence converges to the inductive limit in this new Gromov--Hausdorff propinquity. As an application, we place new strongly Leibniz L-seminorms on AF-algebras using Frobenius--Rieffel norms, for which we have convergence of the Effros--Shen algebras in the Gromov--Hausdorff propinquity with respect to their irrational parameter.
In the context of metric geometry, we introduce a new necessary and sufficient condition for the convergence of an inductive sequence of quantum compact metric spaces for the Gromov-Hausdorff propinquity, which is a noncommutative analogue of the Gromov-Hausdorff distance for compact metric spaces. This condition is easy to verify in many examples, such as quantum compact metric spaces associated to AF algebras or certain twisted convolution C*-algebras of discrete inductive limit groups. Our condition also implies the convergence of an inductive sequence of spectral triples in the sense of the spectral propinquity, a generalization of the Gromov-Hausdorff propinquity on quantum compact metric spaces to the space of metric spectral triples. In particular we show the convergence of the state spaces of the underlying C*-algebras as quantum compact metric spaces, and also the convergence of the quantum dynamics induced by the Dirac operators in the spectral triples. We apply these results to new classes of inductive limit of even spectral triples on noncommutative solenoids and Bunce-Deddens C*-algebras. Our construction, which involves length functions with bounded doubling, adds geometric information and highlights the structure of these twisted C*-algebras as inductive limits.
We describe, implement and test a novel method for training neural networks to estimate the Jacobian matrix $J$ of an unknown multivariate function $F$. The training set is constructed from finitely many pairs $(x,F(x))$ and it contains no explicit information about $J$. The loss function for backpropagation is based on linear approximations and on a nearest neighbor search in the sample data. We formally establish an upper bound on the uniform norm of the error, in operator norm, between the estimated Jacobian matrix provided by the algorithm and the actual Jacobian matrix, under natural assumptions on the function, on the training set and on the loss of the neural network during training. The Jacobian matrix of a multivariate function contains a wealth of information about the function and it has numerous applications in science and engineering. The method given here represents a step in moving from black-box approximations of functions by neural networks to approximations that provide some structural information about the function in question.
The spectral propinquity is a distance, up to unitary equivalence, on the class of metric spectral triples. We prove in this paper that if a sequence of metric spectral triples converges for the propinquity, then the spectra of the Dirac operators for these triples do converge to the spectrum of the Dirac operator at the limit. We obtain this result by first proving that, in an appropriate sense induced by some natural metric, the bounded, continuous functional calculi defined by the Dirac operators also converge. As an application of our work, we see, in particular, that action functionals of a wide class of metric spectral triples are continuous for the spectral propinquity, which clearly connects convergence for the spectral propinquity with the applications of noncommutative geometry to mathematical physics. This fact is a consequence of results on the continuity of the multiplicity of eigenvalues of Dirac operators. In particular, we formalize convergence of adjoinable operators of different C*-correspondences, endowed with appropriate quantum metric data.
Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grow to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sense of the spectral propinquity. This provides a formal assertion that indeed, fuzzy tori approximate quantum tori, not only as quantum metric spaces, but as noncommutative differentiable manifolds -- including convergence of the state spaces as metric spaces and of the quantum dynamics generated by the Dirac operators of the spectral triples, in an appropriate sense.
Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure: for instance, the Sierpiński is the limit of finite graphs consisting of various affine images of an equilateral triangle. It is thus natural to ask whether the spectral triples, constructed on a class of fractals called piecewise $C^1$-fractal curves, are indeed limits, in an appropriate sense, of spectral triples on the approximating sets. We answer this question affirmatively in this paper, where we use the spectral propinquity on the class of metric spectral triples, in order to formalize the sought-after convergence of spectral triples. Our results and methods are relevant to the study of analysis on fractals and have potential physical applications.
We show that Bunce-Deddens algebras, which are AT-algebras, are also limits of circle algebras for Rieffel's quantum Gromov-Hausdorff distance, and moreover, form a continuous family indexed by the Baire space. To this end, we endow Bunce-Deddens algebras with a quantum metric structure, a step which requires that we reconcile the constructions of the Latremoliere's Gromov-Hausdorff propinquity and Rieffel's quantum Gromov-Hausdorff distance when working on order-unit quantum metric spaces. This work thus continues the study of the connection between inductive limits and metric limits.
We survey the symmetry preserving properties for the dual propinquity, under natural non-degeneracy and equicontinuity conditions. These properties are best formulated using the notion of the covariant propinquity when the symmetries are encoded via the actions of proper monoids and groups. We explore the issue of convergence of Cauchy sequences for the covariant propinquity, which captures, via a compactness result, the fact that proper monoid actions can pass to the limit for the dual propinquity.
We define a metric on the class of metric spectral triples, which is null exactly between spectral triples with unitary equivalent Dirac operators and *-isomorphic underlying C*-algebras. This metric dominates the propinquity, and thus implies metric convergence of the compact quantum metric spaces induced by metric spectral triples. In the process of our construction, we also introduce the covariant modular propinquity, as a key component for the definition of the spectral propinquity.
The dual modular propinquity is a complete metric, up to full modular quantum isometry, on the class metrical quantum vector bundles, i.e. of Hilbert modules endowed with a type of densely defined norm, called a D-norm, which generalize the operator norm given by a connection on a Riemannian manifold. The dual modular propinquity is weaker than the modular propinquity yet it is complete, which is the main purpose of its introduction.
The covariant Gromov-Hausdorff propinquity is a distance on Lipschitz dynamical systems over quantum compact metric spaces, up to equivariant full quantum isometry. It is built from the dual Gromov-Hausdorff propinquity which, as its classical counterpart, is complete. We prove in this paper several sufficient conditions for convergence of Cauchy sequences for the covariant propinquity and apply it to show that many natural classes of dynamical systems are complete for this metric.
We extend the Gromov-Hausdorff propinquity to a metric on Lipschitz dynamical systems, which are given by strongly continuous actions of proper monoids on quantum compact metric spaces via Lipschitz morphisms. We prove that our resulting metric is zero between two Lipschitz dynamical systems if and only if there exists an equivariant full quantum isometry between. We also present sufficient conditions for Cauchy sequences to converge for our new metric, thus exhibiting certain complete classes of Lipschitz dynamical systems. We apply our work to convergence of the dual actions on fuzzy tori to the dual actions on quantum tori. Our framework is general enough to also allow for the study of the convergence of continuous semigroups of positive linear maps and other actions of proper monoids.
The modular Gromov-Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a continuous family for the modular propinquity.
We prove a compactness result for classes of actions of many small categories on quantum compact metric spaces by Lipschitz linear maps, for the topology of the covariant Gromov-Hausdorff propinquity. In particular, our result applies to actions of proper groups by Lipschitz isomorphisms on quantum compact spaces. Our result provides a first example of a structure which passes to the limit of quantum metric spaces for the propinquity, as well as a new method to construct group actions, including from non-locally compact groups seen as inductive limits of compact groups, on unital C*-algebras. We apply our techniques to obtain some properties of closure of certain classes of {\gQqcms s} for the propinquity.
We prove that all the compact metric spaces are in the closure of the class of full matrix algebras for the quantum Gromov-Hausdorff propinquity. We also show that given an action of a compact metrizable group G on a quasi-Leibniz compact quantum metric space (A,Lip), the function associating any closed subgroup of G group to its fixed point C*-subalgebra in A is continuous from the topology of the Hausdorff distance to the topology induced by the propinquity. Our techniques are inspired from our work on AF algebras as quantum metric spaces, as they are based on the use of various types of conditional expectations.
The modular Gromov-Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a family of metrized quantum vector bundles, as a first step in proving that Heisenberg modules form a continuous family for the modular Gromov-Hausdorff propinquity.
We introduce a metric on Hilbert modules equipped with a generalized form of a differential structure, thus extending Gromov-Hausdorff convergence theory to vector bundles and quantum vector bundles --- not convergence as total space but indeed as quantum vector bundle. Our metric is new even in the classical picture, and creates a framework for the study of the moduli spaces of modules over C*-algebras from a metric perspective. We apply our construction, in particular, to the continuity of Heisenberg modules over quantum $2$-tori.
We characterize Lipschitz morphisms between quantum compact metric spaces as those *-morphisms which preserve the domain of certain noncommutative analogues of Lipschitz seminorms, namely lower semi-continuous Lip-norms. As a corollary, lower semi-continuous Lip-norms with a shared domain are in fact equivalent. We then note that when a family of lower semi-continuous Lip-norms are uniformly equivalent, then they give rise to totally bounded classes of quantum compact metric spaces, and we apply this observation to several examples of perturbations of quantum metric spaces. We also construct the noncommutative generalization of the Lipschitz distance between quantum compact metric spaces.
We prove that noncommutative solenoids are limits, in the sense of the Gromov-Hausdorff propinquity, of quantum tori. From this observation, we prove that noncommutative solenoids can be approximated by finite dimensional quantum compact metric spaces, and that they form a continuous family of quantum compact metric spaces over the space of multipliers of the solenoid, properly metrized.
We construct quantum metric structures on unital AF algebras with a faithful tracial state, and prove that for such metrics, AF algebras are limits of their defining inductive sequences of finite dimensional C*-algebras for the quantum propinquity. We then study the geometry, for the quantum propinquity, of three natural classes of AF algebras equipped with our quantum metrics: the UHF algebras, the Effros-Shen AF algebras associated with continued fraction expansions of irrationals, and the Cantor space, on which our construction recovers traditional ultrametrics. We also exhibit several compact classes of AF algebras for the quantum propinquity and show continuity of our family of Lip-norms on a fixed AF algebra. Our work thus brings AF algebras into the realm of noncommutative metric geometry.
We prove that curved noncommutative tori, introduced by Dabrowski and Sitarz, are Leibniz quantum compact metric spaces and that they form a continuous family over the group of invertible matrices with entries in the commutant of the quantum tori in the regular representation, when this group is endowed with a natural length function.
We present a survey of the dual Gromov-Hausdorff propinquity, a noncommutative analogue of the Gromov-Hausdorff distance which we introduced to provide a framework for the study of the noncommutative metric properties of C*-algebras. We first review the notions of quantum locally compact metric spaces, and present various examples of such structures. We then explain the construction of the dual Gromov-Hausdorff propinquity, first in the context of quasi-Leibniz quantum compact metric spaces, and then in the context of pointed quantum proper metric spaces. We include a few new result concerning perturbations of the metrics on Leibniz quantum compact metric spaces in relation with the dual Gromov-Hausdorff propinquity.
We prove a compactness theorem for the dual Gromov-Hausdorff propinquity as a noncommutative analogue of the Gromov compactness theorem for the Gromov-Hausdorff distance. Our theorem is valid for subclasses of quasi-Leibniz compact quantum metric spaces of the closure of finite dimensional quasi-Leibniz compact quantum metric spaces for the dual propinquity. While finding characterizations of this class proves delicate, we show that all nuclear, quasi-diagonal quasi-Leibniz compact quantum metric spaces are limits of finite dimensional quasi-Leibniz compact quantum metric spaces. This result involves a mild extension of the definition of the dual propinquity to quasi-Leibniz compact quantum metric spaces, which is presented in the first part of this paper.
Let $p\in \mathbb{N}$ be prime, and let $θ$ be irrational. The authors have previously shown that the noncommutative $p$-solenoid corresponding to the multiplier of the group $\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)^2$ parametrized by $α=(θ+1, (θ+1)/p,\cdots, (θ+1)/p^j,\cdots )$ is strongly Morita equivalent to the noncommutative solenoid on $\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)^2$ coming from the multiplier $β= (1-\frac{θ+1}θ,1-\frac{θ+1}{pθ}, \cdots, 1-\frac{θ+1}{p^jθ}, \cdots )$ . The method used a construction of Rieffel referred to as the "Heisenberg bimodule" in which the two noncommutative solenoid corresponds to two different twisted group algebras associated to dual lattices in $(\mathbb Q_p\times \mathbb R)^2.$ In this paper, we make three additional observations: first, that at each stage, the subalgebra given by the irrational rotation algebra corresponding to $α_{2j}=(θ+1)/p^{2j}$ is strongly Morita equivalent to the irrational rotation algebra corresponding to the irrational rotation algebra corresponding to $β_{2j}= 1-\frac{θ+1}{p^{2j}θ}$ by a different construction of Rieffel, secondly, that that Rieffel's Heisenberg module relating the two non commutative solenoids can be constructed as the closure of a nested sequence of function spaces associated to a multiresolution analysis for a $p$-adic wavelet, and finally, at each stage, the equivalence bimodule between $A_{α_{2j}}$ and $A_{β_{2j}}$ can be identified with the subequivalence bimodules arising from the $p$-adic MRA. Aside from its instrinsic interest, we believe this construction will guide us in our efforts to show that certain necessary conditions for two noncommutative solenoids to be strongly Morita equivalent are also sufficient.
We construct a topology on the class of pointed proper quantum metric spaces which generalizes the topology of the Gromov-Hausdorff distance on proper metric spaces, and the topology of the dual propinquity on Leibniz quantum compact metric spaces. A pointed proper quantum metric space is a special type of quantum locally compact metric space whose topography is proper, and with properties modeled on Leibniz quantum compact metric spaces, though they are usually not compact and include all the classical proper metric spaces. Our topology is obtained from an infra-metric which is our analogue of the Gromov-Hausdorff distance, and which is null only between isometrically isomorphic pointed proper quantum metric spaces. Thus, we propose a new framework which extends noncommutative metric geometry, and in particular noncommutative Gromov-Hausdorff topology, to the realm of quantum locally compact metric spaces.
The dual Gromov-Hausdorff propinquity is a generalization of the Gromov-Hausdorff distance to the class of Leibniz quantum compact metric spaces, designed to be well-behaved with respect to C*-algebraic structures. In this paper, we present a variant of the dual propinquity for which the triangle inequality is established without the recourse to the notion of journeys, or finite paths of tunnels. Since the triangle inequality has been a challenge to establish within the setting of Leibniz quantum compact metric spaces for quite some time, and since journeys can be a complicated tool, this new form of the dual propinquity is a significant theoretical and practical improvement. On the other hand, our new metric is equivalent to the dual propinquity, and thus inherits all its properties.
Quantum tori are limits of finite dimensional C*-algebras for the quantum Gromov-Hausdorff propinquity, a metric defined by the author as a strengthening of Rieffel's quantum Gromov-Hausdorff designed to retain the C*-algebraic structure. In this paper, we propose a proof of the continuity of the family of quantum and fuzzy tori which relies on explicit representations of the C*-algebras rather than on more abstract arguments, in a manner which takes full advantage of the notion of bridge defining the quantum propinquity.
Let p be prime. A noncommutative p-solenoid is the C*-algebra of Z[1/p] x Z[1/p] twisted by a multiplier of that group, where Z[1/p] is the additive subgroup of the field Q of rational numbers whose denominators are powers of p. In this paper, we survey our classification of these C*-algebras up to *-isomorphism in terms of the multipliers on Z[1/p], using techniques from noncommutative topology. Our work relies in part on writing these C*-algebras as direct limits of rotation algebras, i.e. twisted group C*-algebras of the group Z^2 thereby providing a mean for computing the K-theory of the noncommutative solenoids, as well as the range of the trace on the K_0 groups. We also establish a necessary and sufficient condition for the simplicity of the noncommutative solenoids. Then, using the computation of the trace on K_0, we discuss two different ways of constructing projective modules over the noncommutative solenoids.
Motivated by the quest for an analogue of the Gromov-Hausdorff distance in noncommutative geometry which is well-behaved with respect to C*-algebraic structures, we propose a complete metric on the class of Leibniz quantum compact metric spaces, named the dual Gromov-Hausdorff propinquity. This metric resolves several important issues raised by recent research in noncommutative metric geometry: it makes *-isomorphism a necessary condition for distance zero, it is well-adapted to Leibniz seminorms, and --- very importantly --- is complete, unlike the quantum propinquity which we introduced earlier. Thus our new metric provides a natural tool for noncommutative metric geometry, designed to allow for the generalizations of techniques from metric geometry to C*-algebra theory.
We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the Gromov-Hausdorff distance to noncommutative geometry and strengthens Rieffel's quantum Gromov-Hausdorff distance and Rieffel's proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*-algebras.
We introduce the notion of a quantum locally compact metric space, which is the noncommutative analogue of a locally compact metric space, and generalize to the nonunital setting the notion of quantum metric spaces introduced by Rieffel. We then provide several examples of such structures, including the Moyal plane, as well as compact quantum metric spaces and locally compact metric spaces. This paper provides an answer to the question raised in the literature about the proper notion of a quantum metric space in the nonunital setup and offers important insights into noncommutative geometry for non compact quantum spaces.
Noncommutative domain algebras are noncommutative analogues of the algebras of holomorphic functions on domains of $\C^n$ defined by holomorphic polynomials, and they generalize the noncommutative Hardy algebras. We present here a complete classification of these algebras based upon techniques inspired by multivariate complex analysis, and more specifically the classification of domains in hermitian spaces up to biholomorphic equivalence.
A noncommutative solenoid is the C*-algebra $C^\ast(\Q_N^2,σ)$ where $\Q_N$ is the group of the $N$-adic rationals twisted and $σ$ is a multiplier of $\Q_N^2$. In this paper, we use techniques from noncommutative topology to classify these C*-algebras up to *-isomorphism in terms of the multipliers of $\Q_N^2$. We also establish a necessary and sufficient condition for simplicity of noncommutative solenoids, compute their K-theory and show that the $K_0$ groups of noncommutative solenoids are given by the extensions of $\Z$ by $\Q_N$. We give a concrete description of non-simple noncommutative solenoids as bundle of matrices over solenoid groups, and we show that irrational noncommutative solenoids are real rank zero AT C*-algebras.
We investigate the structure of the automorphism of $\mathcal{O}_{2}$ which exchanges the two canonical isometries. Our main observation is that the fixed point C*-subalgebra for this action is isomorphic to $\mathcal{O}_{2}$ and we detail the relationship between the crossed-product and fixed point subalgebra.
We establish that particular quotients of the non-commutative Hardy algebras carry ergodic actions of convergent discrete subgroups of the group $\operatorname*{SU}(n,1)$ of automorphisms of the unit ball in $\mathbb{C}% ^{n}$. To do so, we provide a mean to compute the spectra of quotients of noncommutative Hardy algebra and characterize their automorphisms in term of biholomorphic maps of the unit ball in $\mathbb{C}^{n}$.
This paper extends the results of the previous work of the authors on the classification on noncommutative domain algebras up to completely isometric isomorphism. Using Sunada's classification of Reinhardt domains in $C^n$, we show that aspherical noncommutative domain algebras are isomorphic if and only if their defining symbols are equivalent, in the sense that one can be obtained from the other via permutation and scaling of the free variables. Our result also shows that the automorphism groups of aspherical noncommutative domain algebras consists of a subgroup of some finite dimensional unitary group. We conclude by illustrating how our methods can be used to extend to noncommutative domain algebras some results from analysis in $C^n$ with the example of Cartan's lemma.
Bounded orbit injection equivalence is an equivalence relation defined on minimal free Cantor systems which is a candidate to generalize flip Kakutani equivalence to actions of the Abelian free groups on more than one generator. This paper characterizes bounded orbit injection equivalence in terms of a mild strengthening of Rieffel-Morita equivalence of the associated C*-crossed-product algebras. Moreover, we construct an ordered group which is an invariant for bounded orbit injection equivalence, and does not agrees with the K\_0 group of the associated C*-crossed-product in general. This new invariant allows us to find sufficient conditions to strengthen bounded orbit injection equivalence to orbit equivalence and strong orbit equivalence.
This article discusses the concept of Boolean spaces endowed with a Boolean valued inner product and their matrices. A natural inner product structure for the space of Boolean n-tuples is introduced. Stochastic boolean vectors and stochastic and unitary Boolean matrices are studied. A dimension theorem for orthonormal bases of a Boolean space is proven. We characterize the invariant stochastic Boolean vectors for a Boolean stochastic matrix and show that they can be used to reduce a unitary matrix. Finally, we obtain a result on powers of stochastic and unitary matrices.
Noncommutative domain algebras were introduced by Popescu as the non-selfadjoint operator algebras generated by weighted shifts on the Full Fock space. This paper uses results from several complex variables to classify many noncommutative domain algebras, and it uses results from operator theory to obtain new bounded domains in hermitian spaces with non-compact automorphic group.
We present a characterization of the standard sequential product of quantum effects. The characterization is in term of algebraic, continuity and duality conditions that can be physically motivated.
We describe the structure of the irreducible representations of crossed products of unital C*-algebras by actions of finite groups in terms of irreducible representations of the C*-algebras on which the groups act. We then apply this description to derive a characterization of irreducible representations of crossed-products by finite cyclic groups in terms of representations of the C*-algebra and its fixed point subalgebra. These results are applied to crossed-products by the permutation group on three elements and illustrated by various examples.
We describe the representation theory of C*-crossed-products of a unital C*-algebra A by the cyclic group of order 2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of A. We characterize each class in term of the restriction of the representations to the fixed point C*-subalgebra of A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.
We compute the K-theory of the C*-algebra of symmetric words in two universal unitaries. This algebra is the fixed point C*-algebra for the order-two automorphism of the full C*-algebra of the free group on two generators which switches the generators. Our calculations relate the K -theory of this C*-algebra to the K-theory of the associated C*-crossed-product by Z_2.
We study the C*-algebra crossed-product of the closed unit disk by the action of one of its conformal automorphisms. After classifying the conformal automorphisms up to topological conjugacy, we investigate, for each class, the irreducible representations of the full C*-crossed-products, and derive their spectrum and a complete desciption of the algebras.
Metric noncommutative geometry, initiated by Alain Connes, has known some great recent developments under the impulsion of Rieffel and the introduction of the category of compact quantum metric spaces topologized thanks to the quantum Rieffel-Gromov-Hausdorff distance. In this paper, we undertake the first step to generalize such results and constructions to locally compact quantum metric spaces. Our present work shows how to generalize the construction of the bounded-Lipschitz metric on the state space of a C*-algebra which need not be unital, such that the topology induced by this distance on the state space is the weak* topology. In doing so we obtain some results on a state space picture of the strict topology of a C*-algebra.
The statement that one can approximate a quantum torus by some twisted convolution C*-algebra of a (finite) quotient of Z^d can be found in the physics literature dealing with quantum field theory and M-theory. In this paper, we show that indeed the quantum tori are limits of such finite dimensional C*-algebras for the quantum Gromov-Hausdorff distance for compact quantum metric spaces introduced by Rieffel, when one chooses the right metric structures. The proof uses a mild extension of a result by Ramazan on continuous fields of C*-algebras obtained from groupoid C*-algebras, and extends on Rieffel's work on the continuity of the quantum tori for the quantum Gromov-Hausdorff distance. We also show that one can collapse a quantum torus to a lower dimensional quantum torus.