Finite-dimensional Matrix Algebras Research Articles

Data assimilation in operator algebras

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

Complete Entropic Inequalities for Quantum Markov Chains

We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or L_2-estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.

Characterization of Equivariant Maps and Application to Entanglement Detection

We study equivariant linear maps between finite-dimensional matrix algebras, as introduced by Bhat. These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitary conjugation as studied by Bhat and as a generalization of the equivariant maps studied by Collins et al. Using representation theory, we fully compute them and study their graphical representation, and show that they are basically enough to study all equivariant maps. We finally apply them to the problem of entanglement detection and prove that they form a sufficient (infinite) family of positive maps to detect all k-entangled density matrices.

Primary decompositions of unital locally matrix algebras

We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser. 22(64)(3) (1948) 443–454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. We also show that for an arbitrary infinite Steinitz number [Formula: see text] there exists a unital locally matrix algebra [Formula: see text] having the Steinitz number [Formula: see text] and not isomorphic to a tensor product of finite-dimensional matrix algebras.

Sampling and periodization of generators of Heisenberg modules

This paper considers generators of Heisenberg modules in the case of twisted group [Formula: see text]-algebras of closed subgroups of locally compact abelian (LCA) groups and how the restriction and/or periodization of these generators yield generators for other Heisenberg modules. Since generators of Heisenberg modules are exactly the generators of (multi-window) Gabor frames, our methods are going to be from Gabor analysis. In the latter setting, the procedure of restriction and periodization of generators is well known. Our results extend this established part of Gabor analysis to the general setting of LCA groups. We give several concrete examples where we demonstrate some of the consequences of our results. Finally, we show that vector bundles over an irrational noncommutative torus may be approximated by vector bundles for finite-dimensional matrix algebras that converge to the irrational noncommutative torus with respect to the module norm of the generators, where the matrix algebras converge in the quantum Gromov–Hausdorff distance to the irrational noncommutative torus.

On a family of linear maps from [formula omitted] to [formula omitted

Bhat [1] characterizes the family of linear maps defined on B(H) which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study k-positivity of linear maps defined on finite-dimensional matrix algebras.In [5] Choi showed that n-positivity is different from (n−1)-positivity for the linear maps defined on n by n matrix algebras.In this paper, we present a parametric family of linear maps Φα,β,n:Mn(C)→Mn2(C) and study the properties of positivity, completely positivity, decomposability etc. We determine values of parameters α and β for which the family of maps Φα,β,n is positive for any natural number n≥3. We focus on the case of n=3, that is, Φα,β,3 and study the properties of 2-positivity, completely positivity and decomposability. In particular, we give values of parameters α and β for which the family of maps Φα,β,3 is 2-positive and not completely positive.

On maximal non-selfadjoint reflexive algebras associated with a double triangle lattice

We show that the reflexive algebra Alg(ℒ) given by a double triangle lattice ℒ in a finite factor ℳ (with ℒ″ = ℳ) is maximal non-selfadjoint in the class of all weak operator closed subalgebras with the same diagonal subalgebra Alg(ℒ)∩ Alg(ℒ)*. Our method can be used to prove similar results in finite-dimensional matrix algebras. As a consequence, we give a new proof to the main theorem by Hou and Zhang (2012)

Stable Subspaces of Positive Maps of Matrix Algebras

We study stable subspaces of positive extremal maps of finite dimensional matrix algebras that preserve trace and matrix identity (so-called bistochastic maps). We have established the existence of the isometric-sweeping decomposition for such maps. As the main result of the paper, we have shown that all extremal bistochastic maps acting on the algebra of matrices of size 3×3 fall into one of the three possible categories, depending on the form of the stable subspace of the isometric-sweeping decomposition. Our example of an extremal atomic positive map seems to be the first one that handles the case of that subspace being non-trivial. Lastly, we compute the entanglement witness associated with the extremal map and specify a large family of entangled states detected by it.

Hypercontractivity in finite-dimensional matrix algebras

We obtain hypercontractivity estimates for a large class of semigroups defined on finite-dimensional matrix algebras 𝕄n. These semigroups arise from Poisson-like length functions ψ on ℤn × ℤn and provide new hypercontractive families of quantum channels when ψ is conditionally negative. We also study the optimality of our estimates.

A Construction of Finite Index C*-algebra Inclusions from Free Actions of Compact Quantum Groups

Given an action of a compact quantum group on a unital C*-algebra, one can amplify the action with an adjoint representation of the quantum group on a finite dimensional matrix algebra, and consider the resulting inclusion of fixed point algebras. We show that this inclusion is a finite index inclusion of C*-algebras when the quantum group acts freely. We show that two natural definitions for a quantum group to act freely, namely the Ellwood condition and the saturatedness condition, are equivalent.

Co-amenability and Connes’s embedding problem

A longstanding open question of Connes asks whether any finite von Neumann algebra embeds into an ultraproduct of finite-dimensional matrix algebras. As of yet, algebras verified to satisfy the Connes’s embedding property belong to just a few special classes (e.g., amenable algebras and free group factors). In this article, we prove that von Neumann algebras satisfying Popa’s co-amenability have Connes’s embedding property.

Noncommutative Yang–Mills–Higgs actions from derivation-based differential calculus

Derivations of a noncommutative algebra can be used to construct differential calculi, the so-called derivation-based differential calculi. We apply this framework to a version of the Moyal algebra ${\cal{M}}$. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of ${\cal{M}}$ that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang-Mills-Higgs models on ${\cal{M}}$ and a natural interpretation of the covariant coordinates as Higgs fields. We also compare in detail the main mathematical properties characterizing the present situation to those specific of two other noncommutative geometries, namely the finite dimensional matrix algebra $M_n({\mathbb{C}})$ and the algebra of matrix valued functions $C^\infty(M)\otimes M_n({\mathbb{C}})$. The UV/IR mixing problem of the resulting Yang-Mills-Higgs models is also discussed.

FUZZY COMPLEX GRASSMANNIAN SPACES AND THEIR STAR PRODUCTS

We derive an explicit expression for an associative star product on noncommutative versions of complex Grassmannian spaces, in particular for the case of complex two-planes. Our expression is in terms of a finite sum of derivatives. This generalizes previous results for complex projective spaces and gives a discrete approximation for the Grassmannians in terms of a noncommutative algebra, represented by matrix multiplication in a finite-dimensional matrix algebra. The matrices are restricted to have a dimension which is precisely determined by the harmonic expansion of functions on the commutative Grassmannian, truncated at a finite level. In the limit of infinite-dimensional matrices we recover the commutative algebra of functions on the complex Grassmannians.

A COUNTEREXAMPLE TO A CONJECTURE OF AKEMANN AND ANDERSON

Akemann and Anderson made a conjecture about ``paving'' projections in finite dimensional matrix algebras which, if true, would settle the well-known Kadison-Singer problem. We falsify their conjecture by an explicit seqence of counterexamples.

A REMARK ON THE STRUCTURE OF SYMMETRIC QUANTUM DYNAMICAL SEMIGROUPS ON VON NEUMANN ALGEBRAS

We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with Abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results obtained by Albeverio, Høegh-Krohn and Olsen1 for the special case of the finite-dimensional matrix algebras. We also study similar questions for a class of quantum dynamical semigroups with unbounded generators.

Gravity, non-commutative geometry and the Wodzicki residue

We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator D on an n dimensional compact Riemannian manifold with n ≥ 4, n even, the Wodzicki residue Res( D − n+2 ) is the integral of the second coefficient of the heat kernel expansion of D 2. We use this result to derive a gravity action for commutative geometry which is the usual Einstein-Hilbert action and we also apply our results to a non-commutative extension which is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.

HIGHER SPIN ALGEBRAS AND QUANTIZATION ON THE SPHERE AND HYPERBOLOID

The oscillator-type realization is proposed for the continuous set of infinite-dimensional algebras of quantum operators on the two-dimensional sphere and hyperboloid. This realization is typical for infinite-dimensional higher spin algebras related to higher spin gauge theories. It involves the Klein-type operator that emerges nontrivially in the Heisenberg-type commutation relations for the oscillators. The invariant trace and bilinear form are constructed. The latter is shown to degenerate for all odd-integer values of the continuous parameter ν, which parametrizes the class of algebras under investigation. The degeneration points are shown to correspond to ordinary finite-dimensional matrix algebras and superalgebras. Possible applications of these results to higher spin gauge theories are discussed. In particular, it is noted that the deformation parameter ν can be interpreted as a vacuum value of some auxiliary scalar field in an appropriate higher spin gauge theory.

A Finite But Not Stably Finite C ∗ -Algebra

In E. G. Effros's paper [4] given at the 1980 Kingston Conference of the American Mathematical Society, he stated that example of a finite but not stably finite C*-algebra has yet to be found. This paper seeks to give an example of such an algebra by using a simple application of the duality between Kand Ext-theory arising from the work of Brown, Douglas and Fillmore (see, for example, [2 and 3]). Throughout this paper the C*-algebras will be separable and unital, except in the obvious cases such as X, the algebra of compact operators on an infinite-dimensional, separable Hilbert space XV. 2 will denote the algebra of all bounded, linear operators on JX, so that X is the only ideal (= closed, two-sided ideal) of Y. The quotient is 12, the Calkin algebra. Denote by U( -V) the group of all unitary operators in Y. Finally, for each positive integer n let M,1 be the algebra of complex valued, n x n matrices. Let A be a separable, unital C *-algebra. We say that A is finite if for every pair of elements x, y in A, xy= 1 implies yx= 1. A is stably finite if every finite-dimensional matrix algebra over A is finite; that is, if Mn 0 A is finite for each positive integer n. It is now time to introduce the notion of K-theory. Again, let A be a separable, unital C*-algebra. We can define a semigroup J(A) as the set of projections in YX A with direct sum as addition, modulo equivalence of projections. Following [6] this equivalence of projections can be defined in three different ways (equivalence, *-equivalence and unitary equivalence), but as Kaplansky shows [6, Chapter 3, in particular Theorem 28], these all give the same relation on a stable algebra. Alternatively, J(A) can be defined as equivalence classes of idempotents in )r A with direct sum as addition. Again, the two relevant types of equivalence (equivalence and invertible equivalence) turn out to be the same, and this version of J(A) is the same as the last [6, Theorem 27]. For e an idempotent in )r A, denote by [e]j the class of e in J(A). Following J. L. Taylor [9, Chapter 6], let KO(A) be the Received by the editors Jaunary 11, 1984 and, in revised form, October 3, 1984. 1980 Mathematics Subject Classificcation. Primary 46L05.

A Construction of Approximately Finite-Dimensional Non-ITPFI Factors

A von Neumann algebra is said to be approximately finite-dimensional if it is of the formwhere Mn⊆Mn+1 for each n and each Mn is a finite-dimensional matrix algebra. A factor is said to be ITPFI if it is of the form

Numerical invariants and projective modules

In a recent paper [9], Fr6hlich gives criteria in terms of numerical invariants to insure the projectivity of modules of an order in a commutative separable algebra over the quotient field of a Dedekind domain. More precisely, Frijhlich first defines a numerical invariant, the module index (see [8] for the definition), and then considers the following context: let A be a Dedekind domain, K its quotient field, 2 a finite dimensional commutative separable K-algebra, r the unique maximal A-order in Z and A any other A-order of Z. Now if M is a A-module such that M @A K E LY’), for an integer I, then M is A-projective if and only if [FM : M] = [r : A]’ (I’M is taken to denote the smallest r-module containing M, and [P : Q] is the notation for the module index of A-lattices P and Q). For purposes of reference we call this result “Friihlich’s theorem.” The object of this paper is to extend and give analogues to FrBhlich’s results for the case that Z is a finite dimensional separable K-algebra. It will be seen, however, that in the fairly simple case of a finite dimensional matrix algebra over the quotient field of a (complete) discrete rank one valuation rank, no direct analogue of either the necessity or the sufficiency of Fr6hlich’s theorem can be given. Nevertheless, for any finite dimensional central simple algebra Z over the quotient field of a complete discrete rank one valuation ring, we can give results similar to those of Frijhlich for some special types of A-orders in 2. Further, we shall show that Fr6hlich’s results extend almost entirely when Z is a central division K-algebra and the base ring A is a complete discrete rank one valuation ring. FrGhlich also gave a criteria to determine the projectivity of fractional ideals I in A such that I BA K E 2. This can be stated as: I is A-projective