Dense Subalgebra 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.

The noncommutative geometry of the Landau Hamiltonian: differential aspects

In this work we study the differential aspects of the noncommutative geometry for the magnetic C*-algebra which is a 2-cocycle deformation of the group C*-algebra of . This algebra is intimately related to the study of the quantum Hall effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo’s formula. Taking inspiration from the ideas developed by Bellissard during the 80s, we build an appropriate Fredholm module for the magnetic C*-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard’s theory, the so-called second Connes’ formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic C*-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes’ quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

Atom canonicity and first order definability in classes of algebras of relations

AbstractFix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of Ln. Members of RCAn are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever 𝔄 is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated Henkin-Orey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of Lω,ω, fails dramatically for Ln even if we allow certain generalized models that are only locallly clasfsical. It is also shown that any class K such that , where CRCAn is the class of completely representable CAns, and Sc denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that is not elementary, where Sd denotes the operation of forming dense subalgebra.

Multiplication operators on Hardy and weighted Bergman spaces over planar regions

This paper studies some aspects of commutant theory and functional calculus for analytic multiplication operators on Hardy and weighted Bergman spaces over bounded planar regions. Multiplication operators defined by univalent functions are shown to commute only with multiplication operators. This result is generalized to a tuple of operators, and a sufficient condition is given for irreducibility of that induced by finite Blaschke products. Operators defined by fairly general ancestral functions are shown to commute with no nonzero compact operators, and these include the ones by monomial functions over annuli. For such operators over annuli, we characterize a certain dense subalgebra of the commutant. Norm and sequential weak closures of the analytic functional calculus algebra generated by a multiplication operator are characterized and essential spectral mapping properties obtained. Generalizing the similarity for finite Blaschke products to a larger class of weighted Bergman spaces, the commutant classification of these operators is obtained and seen strictly finer than the similarity classification, among other related results.

Point-evaluation functionals on algebras of symmetric functions on $(L_\infty)^2$

It is known that every continuous symmetric (invariant under the composition of its argument with each Lebesgue measurable bijection of $[0,1]$ that preserve the Lebesgue measure of measurable sets) polynomial on the Cartesian power of the complex Banach space $L_\infty$ of all Lebesgue measurable essentially bounded complex-valued functions on $[0,1]$ can be uniquely represented as an algebraic combination, i.e., a linear combination of products, of the so-called elementary symmetric polynomials. Consequently, every continuous complex-valued linear multiplicative functional (character) of an arbitrary topological algebra of the functions on the Cartesian power of $L_\infty,$ which contains the algebra of continuous symmetric polynomials on the Cartesian power of $L_\infty$ as a dense subalgebra, is uniquely determined by its values on elementary symmetric polynomials. Therefore, the problem of the description of the spectrum (the set of all characters) of such an algebra is equivalent to the problem of the description of sets of the above-mentioned values of characters on elementary symmetric polynomials.
 In this work, the problem of the description of sets of values of characters, which are point-evaluation functionals, on elementary symmetric polynomials on the Cartesian square of $L_\infty$ is completely solved. We show that sets of values of point-evaluation functionals on elementary symmetric polynomials satisfy some natural condition. Also, we show that for any set $c$ of complex numbers, which satisfies the above-mentioned condition, there exists an element $x$ of the Cartesian square of $L_\infty$ such that values of the point-evaluation functional at $x$ on elementary symmetric polynomials coincide with the respective elements of the set $c.$

On the smoothness of some quotients of Banach-Lie groups

Quotients of Banach-Lie groups are regarded as topological groups with Lie algebra in the sense of Hofmann-Morris on the one hand, and as Q-groups in the sense of Barre-Plaisant on the other hand. For the groups of the type G/N where N⊆G is a pseudo-discrete normal subgroup, their Lie algebra in the sense of Q-groups turns out to be isomorphic to the Lie algebra of G, which is in general merely a dense subalgebra of the Lie algebra of G/N when regarded as a topological group with Lie algebra. The submersion-like behavior of quotient maps of Banach-Lie groups is also investigated. The two aforementioned approaches to the Lie theory of the quotients of Banach-Lie groups thus lead to differing results and the Lie theoretic properties of quotient groups are more accurately described by the Q-group approach than by the approach via topological groups with Lie algebras.

Noncommutative Kähler structure on [formula omitted]-dynamical systems

Notions of noncommutative complex and Kähler structure have been introduced by Fröhlich et al. (1999), in the context of supersymmetric quantum theory. Here we show that whenever a C∗-dynamical system (A,G,α,τ) equipped with a faithful G-invariant trace τ, where G is an even dimensional abelian Lie group, determines a spectral triple, the smooth dense subalgebra A∞ inherits a noncommutative Kähler structure. In particular, whenever T2n acts ergodically on the algebra, it inherits a noncommutative Kähler structure. This produces a class of examples of noncommutative Kähler manifolds. As a corollary, we obtain that all the noncommutative even dimensional tori are noncommutative Kähler manifolds. We explicitly compute the space of complex differential forms and study holomorphic vector bundles on all noncommutative even dimensional tori.

Quantization and the resolvent algebra

We introduce a novel commutative C*-algebra CR(X) of functions on a symplectic vector space (X,σ) admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of CR(X) to the resolvent algebra introduced by Buchholz and Grundling [2]. The associated quantization map is a field-theoretical Weyl quantization compatible with the work of Binz, Honegger and Rieckers [1]. We also define a Berezin-type quantization map on all of CR(X), which continuously and bijectively maps it onto the resolvent algebra.The commutative resolvent algebra CR(X), generally defined on a real inner product space X, intimately depends on the finite dimensional subspaces of X. We thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum.

Lp-operator algebras associated with oriented graphs

For each 1⩽p<∞ and each countable oriented graph Q we introduce an Lp-operator algebra Op(Q), which contains the Leavitt path C-algebra LQ as a dense subalgebra, and is universal for those Lp-representations of LQ which are spatial in the sense of N.C. Phillips. We prove that Op(Q) is simple as an Lp-operator algebra if and only if LQ is simple, in which case it is isometrically isomorphic to ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ρ(LQ) for any nonzero spatial Lp-representation ρ:LQ→L(Lp(X)). If moreover LQ is purely infinite simple and p≠p′, then there is no nonzero continuous homomorphism Op(Q)→Op′(Q).

The Resolvent Algebra of Non-relativistic Bose Fields: Observables, Dynamics and States

The structure of the gauge invariant (particle number preserving) C*-algebra generated by the resolvents of a non-relativistic Bose field is analyzed. It is shown to form a dense subalgebra of the bounded inverse limit of a system of approximately finite dimensional C*-algebras. Based on this observation, it is proven that the closure of the gauge invariant algebra is stable under the dynamics induced by Hamiltonians involving pair potentials. These facts allow to proceed to a description of interacting Bosons in terms of C*-dynamical systems. It is outlined how the present approach leads to simplifications in the construction of infinite bosonic states and sheds new light on topics in many body theory.

Partial actions of $C^{*}$ -quantum groups

Partial actions of groups on C*-algebras and the closely related actions and coactions of Hopf algebras received much attention over the last decades. They arise naturally as restrictions of their global counterparts to non-invariant subalgebras, and the ambient eveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of C*-bialgebras, focussing on C*-quantum, and prove existence of an enveloping global coaction under mild technical assumptions. The construction of the latter provides a left adjoint to the forgetful functor from coactions to partial coactions. We also show that partial coactions of the function algebra of a discrete group correspond to partial actions on direct summands of a C*-algebra, and relate partial coactions of a compact or its dual discrete C*-quantum group to partial coactions or partial actions of the dense Hopf subalgebra.

The Resolvent Algebra for Oscillating Lattice Systems: Dynamics, Ground and Equilibrium States

Within the C*-algebraic framework of the resolvent algebra for canonical quantum systems, the structure of oscillating lattice systems with bounded nearest neighbor interactions is studied in any number of dimensions. The global dynamics of such systems acts on the resolvent algebra by automorphisms and there exists a (in any regular representation) weakly dense subalgebra on which this action is pointwise norm continuous. Based on this observation, equilibrium (KMS) states as well as ground states are constructed which are shown to be regular. It is also indicated how to deal with singular interactions and non-harmonic oscillations.

Finitely-axiomatizable superatomic Boolean algebras with distinguished dense subalgebra of finite width

We give a description of finitely axiomatizable superatomic Boolean algebras with distinguished dense subalgebra of finite width. Criteria are obtained for the elementary equivalence of superatomic Boolean algebras with distinguished dense subalgebra of finite width and the decidability of their elementary theories.

K-homological finiteness and hyperbolic groups

Abstract Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C * \mathrm{C}^{*} -algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of C * \mathrm{C}^{*} -algebras associated to hyperbolic groups – the C * \mathrm{C}^{*} -crossed product for the boundary action, and the reduced group C * \mathrm{C}^{*} -algebra – have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.

Smooth Frechet subalgebras of C ∗ -algebras defined by first order differential seminorms

The differential structure in a C ∗-algebra defined by a dense Frechet subalgebra whose topology is defined by a sequence of differential seminorms of order 1 is investigated. This includes differential Arens–Michael decomposition, spectral invariance, closure under functional calculi as well as intrinsic spectral description. A large number of examples of such Frechet algebras are exhibited; and the smooth structure defined by an unbounded self-adjoint Hilbert space operator is discussed.

A note on latticeability and algebrability

Suppose A is a subset of a Banach lattice (Banach algebra) X. We look for “large” sublattices (resp. subalgebras) of A. If X is a Banach lattice, we prove: (1) If Y is a closed subspace of X of codimension at least n, then (X\\Y)∪{0} contains a sublattice of dimension n. (2) If Y is a closed infinite codimensional ideal in X, then (X\\Y)∪{0} contains a closed infinite dimensional sublattice. (3) If the order in X is induced by a 1-unconditional basis, and Y is a closed infinite codimensional subspace of X, then (X\\Y)∪{0} contains a closed infinite dimensional ideal. Further, we show that (4) (ℓp\\(∪q<pℓq))∪{0} contains a sublattice which is dense in ℓp, and that (5) the sets L1(T)\\(∪p>1Lp(T))∪{0} and S∞\\(∪p<∞Sp)∪{0} contain a dense subalgebra with a continuum of free generators (here Sp denotes the Schatten p-space).

Weak amenability for Fourier algebras of 1-connected nilpotent Lie groups

A special case of a conjecture raised by Forrest and Runde (2005) [10] asserts that the Fourier algebra of every non-abelian connected Lie group fails to be weakly amenable; this was already known to hold in the non-abelian compact cases, by earlier work of Johnson (1994) [13] and Plymen (unpublished note). In recent work (Choi and Ghandehari, 2014 [4]) the authors verified this conjecture for the real ax+b group and hence, by structure theory, for any semisimple Lie group.In this paper we verify the conjecture for all 1-connected, non-abelian nilpotent Lie groups, by reducing the problem to the case of the Heisenberg group. As in our previous paper, an explicit non-zero derivation is constructed on a dense subalgebra, and then shown to be bounded using harmonic analysis. En route we use the known fusion rules for Schrödinger representations to give a concrete realization of the “dual convolution” for this group as a kind of twisted, operator-valued convolution. We also give some partial results for solvable groups which give further evidence to support the general conjecture.

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

Truncation Approximations and Spectral Invariant Subalgebras in Uniform Roe Algebras of Discrete Groups

In this paper we study band truncation approximations for operators in uniform Roe algebras of countable discrete groups. Under conditions on certain growth rates for discrete groups, we find large classes of dense subspaces of uniform Roe algebras whose elements can be approximated by their band truncations in the operator norm. We apply these results to construct a nested family of spectral invariant Banach algebras on discrete groups. For a group with polynomial growth, the intersection of these Banach algebras is a spectral invariant dense subalgebra of the uniform Roe algebra. For a group with subexponential growth, we show that the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra.

On the Structure of Commutative Banach Algebras Generated by Toeplitz Operators on the Unit Ball. Quasi-Elliptic Case. II: Gelfand Theory

Extending our results in Bauer and Vasilevski (J Funct Anal 265(11):2956–2990, 2013) the present paper gives a detailed structural analysis of a class of commutative Banach algebras \(\mathcal {B}_k(h)\) generated by Toeplitz operators on the standard weighted Bergman spaces \(\mathcal {A}_{\lambda }^2(\mathbb {B}^n)\) over the complex unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\). In the most general situation we explicitly determine the set of maximal ideals of \(\mathcal {B}_k(h)\) and we describe the Gelfand transform on a dense subalgebra. As an application to the spectral theory we prove the inverse closedness of algebras \(\mathcal {B}_k(h)\) in the full algebra of bounded operators on \(\mathcal {A}_{\lambda }^2(\mathbb {B}^n)\) for certain choices of \(h\). Moreover, it is remarked that \(\mathcal {B}_k(h)\) is not semi-simple. In the case of \(k=(n)\) we explicitly describe the radical \(\hbox {Rad}\, \mathcal {B}_n(h)\) of the algebra \(\mathcal {B}_n(h)\). This result generalizes and simplifies the characterization of \(\hbox {Rad}\,\mathcal {B}_2(1)\), which was given in Bauer and Vasilevski (Integr Equ Oper Theory 74:199–231, 2012).