Rotation Algebras Research Articles

Spectral triples for noncommutative solenoids and a Wiener’s lemma

In this paper, we construct odd finitely summable spectral triples based on length functions of bounded doubling on noncommutative solenoids. Our spectral triples induce a Leibniz Lip-norm on the state spaces of the noncommutative solenoids, giving them the structure of Leibniz quantum compact metric spaces. By applying methods of R. Floricel and A. Ghorbanpour, we also show that our odd spectral triples on noncommutative solenoids can be considered as inductive limits of spectral triples on rotation algebras. In the final section, we prove a noncommutative version of Wiener’s lemma and show that our odd spectral triples can be defined to have an associated smooth dense subalgebra which is stable under the holomorphic functional calculus, thus answering a question of B. Long and W. Wu. The construction of the smooth subalgebra also extends to the case of nilpotent discrete groups.

R⁎-algebras and C⁎-uniqueness of pre-C⁎-algebras

In 1967, Kadison raised the following question, if a self-adjoint operator algebra on a Hilbert space is finitely generated (algebraically) and each self-adjoint operator in it has finite spectrum, is it finite dimensional? In 2021, M. Mori connected the Kadison's question to von Neumann's concept of regular rings given in 1930's, and introduced a notion of R⁎-algebras. In this paper, we study R⁎-algebras, the Kadison's question, and a related question of uniqueness of C⁎-norm on pre-C⁎-algebras. Precisely, we study atomic R⁎-algebras, and find a maximal ultramatricial R⁎-subalgebra of an atomic R⁎-algebra, which properly contains the maximal purely atomic R⁎-subalgebra generated by all finite projections of the algebra. We prove that the set of all ⁎-algebras for which the Kadison's question has an affirmative answer is closed under direct sum, tensor product, and crossed product, but it is not closed under free product. From the aforementioned stable properties of the algebra in the Kadison's question, we find plentiful examples of ⁎-algebras for which the Kadison's question has an affirmative answer, for instance, the crossed product ⁎-algebra B⋊αG, where G is a locally finite group and B is one of the following algebras: an algebra of finite rank operators, an abelian algebra, a finite dimensional algebra, a twisted group algebra satisfying a certain condition, a rotation algebra, or a group measure space construction algebra. We characterize uniqueness of C⁎-norm on a pre-C⁎-algebras in terms of its ⁎-representations, weak containment of its ⁎-representations, ideals of its enveloping C⁎-algebra, or the primitive ideal space of the C⁎-algebra. We also prove that a unital pre-C⁎-algebra has a unique C⁎-norm if and only if, for a faithful ⁎-representation of the algebra, invertibility of a self-adjoint element of the algebra, as an operator on the representation Hilbert space, is independent of the choice of the ⁎-representation. These characterizations provide answers to the question raised by M. Mori in 2021 on finding a sufficient and necessary condition for a ⁎-algebra to have a unique C⁎-norm. For discrete groups, we give a characterization for a group algebra to have a unique C⁎-norm in terms of the group's unitary representations, providing an answer to the question raised by Alekseev in 2019 on characterizing C⁎-unique discrete groups.

K-theory of noncommutative Bernoulli shifts

For a large class of C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras A, we calculate the K-theory of reduced crossed products A⊗G⋊rG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{\\otimes G}\\rtimes _rG$$\\end{document} of Bernoulli shifts by groups satisfying the Baum–Connes conjecture. In particular, we give explicit formulas for finite-dimensional C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the K-theory of reduced C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras of wreath products H≀G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\wr G$$\\end{document} for large classes of groups H and G. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.

A Balian–Low type theorem for Gabor Riesz sequences of arbitrary density

Gabor systems are used in fields ranging from audio processing to digital communication. Such a Gabor system (g,varLambda ) consists of all time-frequency shifts pi (lambda ) g of a window function g in L^2({mathbb {R}}) along a lattice varLambda subset {mathbb {R}}^2. We focus on Gabor systems that are also Riesz sequences, meaning that one can stably reconstruct the coefficients c = (c_lambda )_{lambda in varLambda } from the function sum _{lambda in varLambda } c_lambda , pi (lambda ) g. In digital communication, a function of this form is used to transmit the digital sequence c. It is desirable for g to be well localized in time and frequency, since the transmitted signal will then be almost compactly supported in time and frequency if the sequence c has finite support. In this paper, we study what additional structural properties the signal space mathcal {G}(g,varLambda ), i.e., the span of the Gabor system, satisfies in addition to being a closed subspace of L^2({mathbb {R}}). The most well-known result in this direction—the Balian–Low theorem—states that if g is well localized in time and frequency and if (g,varLambda ) is a Riesz sequence, then mathcal {G}(g,varLambda ) is necessarily a proper subspace of L^2({mathbb {R}}). We prove a generalization of this result related to the invariance of mathcal {G}(g,varLambda ) under time-frequency shifts. Precisely, we show that if (g,varLambda ) is a Riesz sequence with g being well localized in time and frequency (precisely, g should belong to the so-called Feichtinger algebra), then pi (mu ) mathcal {G}(g,varLambda ) subset mathcal {G}(g,varLambda ) holds if and only if mu in varLambda . For lattices of rational density, this was already known, with the proof based on Zak transform techniques. These methods do not generalize to arbitrary lattices, however. Instead, our proof for lattices of irrational density relies on combining methods from time-frequency analysis with properties of a special C^*-algebra, the so-called irrational rotation algebra.

Homotopies of Constant Cuntz Class

Abstract Let $A$ be a unital simple separable exact C$^{\ast }$-algebra that is approximately divisible and of real rank zero. We prove that the set of positive elements in $A$ with a fixed Cuntz class is path connected. This result applies in particular to irrational rotation algebras and approximately finite-dimensional (AF) algebras.

Completely Compact Herz–Schur Multipliers of Dynamical Systems

We prove that if G is a discrete group and (A,G,alpha ) is a C*-dynamical system such that the reduced crossed product Artimes _{r,alpha } G possesses property (SOAP) then every completely compact Herz–Schur (A,G,alpha )-multiplier can be approximated in the completely bounded norm by Herz–Schur (A,G,alpha )-multipliers of finite rank. As a consequence, if G has the approximation property (AP) then the completely compact Herz–Schur multipliers of A(G) coincide with the closure of A(G) in the completely bounded multiplier norm. We study the class of invariant completely compact Herz–Schur multipliers of Artimes _{r,alpha } G and provide a description of this class in the case of the irrational rotation algebra.

Constructing equivalence bimodules between noncommutative solenoids: A two-pronged approach

We revisit and generalize the application of a method introduced by Latrémolière and Packer for constructing finitely generated projective modules over the noncommutative solenoid C*-algebras. By realizing them as direct limits of rotation algebras, the method constructs directed systems of equivalence bimodules between rotation algebras that satisfy the necessary compatibility conditions to build Morita equivalence bimodules between the direct limit C*-algebras. In the irrational case, we use a fixed projection in a matrix algebra over the rotation algebra satisfying a key condition to build an equivalence bimodule at each stage following a construction of Rieffel. From this, our main result shows that two irrational noncommutative solenoids are Morita equivalent if and only if such a projection exists. We also make additional observations about the Heisenberg bimodules construction studied by the aforementioned two authors and connect the two constructions.

Singular integrals in quantum Euclidean spaces

We shall establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes’ pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduceLpL_p-boundedness and Sobolevpp-estimates for regular, exotic and forbidden symbols in the expected ranks. In theL2L_2level both Calderón-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we proveLpL_p-regularity of solutions for elliptic PDEs.

Lie polynomials in an algebra defined by a linearly twisted commutation relation

We present an elementary approach to characterizing Lie polynomials on the generators [Formula: see text] of an algebra with a defining relation in the form of a twisted commutation relation [Formula: see text]. Here, the twisting map [Formula: see text] is a linear polynomial with a slope parameter, which is not a root of unity. The class of algebras defined as such encompasses [Formula: see text]-deformed Heisenberg algebras, rotation algebras, and some types of [Formula: see text]-oscillator algebras, the deformation parameters of which, are not roots of unity. Thus, we have a general solution for the Lie polynomial characterization problem for these algebras.

A note on crossed products of rotation algebras

We compute the K-theory of crossed products of rotation algebras Aθ, for any real angle θ, by matrices in SL(2,Z) with infinite order. Using techniques of continuous fields, we show that the canonical inclusion of Aθ into the crossed products is injective at the level of K0-groups. We then give an explicit set of generators for the K0-groups and compute the tracial ranges concretely.

Алгоритм калибровки группы MIMU датчиков

Non-optical wearable sensors, such as magnetic and inertial measuring units (MIMU), are becoming popular in various fields: sports, medical, industrial - due to their ease of use and relative availability. We propose an algorithm for calibrating wearable sensors based on the rotation algebra. A system for visualizing human kinematics, which is reconstructed from MIMUs' data, is presented.

Dilations of q-Commuting Unitaries

Abstract Let $q = e^{i \theta } \in \mathbb{T}$ (where $\theta \in \mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, that is, $u$ and $v$ are unitaries such that $vu = quv$. In this paper, we find the optimal constant $c = c_{\theta }$ such that $u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are commuting unitaries. We show that $$\begin{equation*} c_{\theta} = \frac{4}{\|u_{\theta}+u_{\theta}^*+v_{\theta}+v_{\theta}^*\|}, \end{equation*}$$where $u_{\theta }, v_{\theta }$ are the universal $q$-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating $q$-commuting unitaries to scalar multiples of $q^{\prime}$-commuting unitaries. The techniques that we develop allow us to give new and simple “dilation theoretic” proofs of well-known results regarding the continuity of the field of rotations algebras. In particular, for the so-called “almost Mathieu operator” $h_{\theta } = u_{\theta }+u_{\theta }^*+v_{\theta }+v_{\theta }^*$, we recover the fact that the norm $\|h_{\theta }\|$ is a Lipschitz continuous function of $\theta $, as well as the result that the spectrum $\sigma (h_{\theta })$ is a $\frac{1}{2}$-Hölder continuous function in $\theta $ with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint *-polynomial $p(u_{\theta },v_{\theta })$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.

Harmonic Analysis Approach to Gromov–Hausdorff Convergence for Noncommutative Tori

We show that the rotation algebras are limit of matrix algebras in a very strong sense of convergence for algebras with additional Lipschitz structure. Our results generalize to higher dimensional noncommutative tori and operator valued coefficients. In contrast to previous results by Rieffel, Li, Kerr, and Latr\'emoli\`ere we use Lipschitz norms induced by the `carr\'e du champ' of certain natural dynamical systems, including the heat semigroup.

Semiclassical asymptotics of the spectrum near the lower boundary of spectral clusters for a Hartree-type operator

The eigenvalue problem for a perturbed two-dimensional resonant oscillator is considered. The exciting potential is given by a nonlocal nonlinearity of Hartree type with smooth self-action potential. To each representation of the rotation algebra corresponds the spectral cluster around an energy level of the unperturbed operator. Asymptotic eigenvalues and asymptotic eigenfunctions close to the lower boundary of spectral clusters are obtained. For their calculation, asymptotic formulas for quantum means are used.

Mutual embeddability equivalence relation for rotation algebras

Mutual embeddability is an equivalence relation in C⁎-algebras. In this paper, we characterize the classification of rotation algebras in the sense of mutual embeddability.

Open-Loop End-to-End Transmission for Multihop Opportunistic Networks With Energy-Harvesting Devices

Networks formed by energy-harvesting devices impose new technological challenges on data transmission due to uncertainty of the amount of energy that can be harvested. This is even more challenging with multihop networks, in which transmission outage occurs when one single device along the relay path cannot harvest enough energy. A virtual multiple-input multiple-output (MIMO) model for multihop, multipath networks has recently been developed to facilitate reliable open-loop end-to-end transmissions, making networks robust to random transmission outages without the necessity of bandwidth-consuming and complicated end-to-end feedback and control. In this paper, a framework based on a virtual MIMO model of multihop, multipath opportunistic networks formed by energy-harvesting devices is developed. We propose rotated-algebraic path-time codes (RA-PTC), by which data are encoded using Givens rotation and cyclic division algebras. Without rate loss, a form of time diversity is exploited by repeatedly transmitting the RA-PTC-coded data. Extensive theoretical analyses are carried out using amount of fading and diversity as metrics. Both the performance enhancement due to the proposed RA-PTC and the performance loss due to energy shortage are quantified. Furthermore, a simple yet effective cyclic power control is proposed to improve transmission reliability. Numerical results demonstrate that RA-PTC and cyclic power control enable efficient and reliable end-to-end transmission in multipath, multihop opportunistic networks formed by energy-harvesting devices.

All Irrational Extended Rotation Algebras are AF Algebras

AbstractLet θ ∊ [0, 1] be any irrational number. It is shown that the extended rotation algebra Bθ introduced by the authors in J. Reine Angew. Math. 665(2012), pp. 1–71, is always an AF algebra.

Rotation Algebras and the Exel Trace Formula

AbstractWe show that ifuandvare any two unitaries in a unitalC*–algebra such that ∥uv−vu∥ < 2 anduvu*v* commutes withuandv, then theC*–subalgebraAu,vgenerated by u and v is isomorphic to a quotient of some rotation algebraAθ, provided thatAu;vhas a unique tracial state. We also show that the Exel trace formula holds in any unitalC*–algebra. Let θ ∊ (−1/2, 1/2) be a real number. For any ∊ > 0; we prove that there exists ζ > 0 satisfying the following: if u and v are two unitaries in any unital simpleC*–algebra A with tracial rank zero such thatfor all tracial states τof A; then there exists a pair of unitariesandin A such that

An algebraic construction of quantum flows with unbounded generators

It is shown how to construct -homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on C algebras; this generalises the construction of a classical Feller process and semigroup from a given generator. Our construction is possible provided the generator satises an invariance property for some dense subalgebra A0 of the C algebraA and obeys the necessary structure relations; the iterates of the generator, when applied to a generating set for A0, must satisfy a growth condition. Furthermore, it is assumed that either the subalgebra A0 is generated by isometries and A is universal, or A0 contains its square roots. These conditions are veried in four cases: classical random walks on discrete groups, Rebolledo’s symmetric quantum exclusion process and ows on the non-commutative torus and the universal rotation algebra. Des cocycles de Feller stochastiques quantiques -homomorphes sont construits pour certains g en erateurs non born es, et ainsi nous obtenons des dilatations pour des semigroupes

Finite groups acting on higher dimensional noncommutative tori

For the canonical action α of SL2(Z) on 2-dimensional simple rotation algebras Aθ, it is known that if F is a finite subgroup of SL2(Z), the crossed products Aθ⋊αF are all AF algebras. In this paper we show that this is not the case for higher dimensional noncommutative tori. More precisely, we show that for each n≥3 there exist noncommutative simple ϕ(n)-dimensional tori AΘ which admit canonical action of Zn and for each odd n≥7 with 2ϕ(n)≥n+5 their crossed products AΘ⋊αZn are not AF (with nonzero K1-groups). It is also shown that the only possible canonical action by a finite group on a 3-dimensional simple torus is the flip action by Z2. Besides, we discuss the canonical actions by finite groups Z5,Z8,Z10, and Z12 on the 4-dimensional torus of the form Aθ⊗Aθ.