Ergodic actions and spectral triples

The unbounded Self-adjoint operators that strongly commute on a common dense subset of their domain commute pointwise. When the operators commute pointwise on the same dense subset, there is to guarantee that they will commute strongly. By imposing some conditions, we on the operators as well as the underlying space, we get pointwise commuting unbounded operators that commute strongly. This article shows that by suitably selecting two unbounded positive Self-adjoint operators with compact inverses we get a set of pointwise commuting self-adjoint operators that commute on common core. then prove that it strongly commutes on the same subspace. Keywords: Unbounded operators, Self-adjoint operators, Commutative operators DOI: 10.7176/MTM/10-8-03 Publication date: December 31 st 2020

Quantum double suspension and spectral triples

Metric approximations of spectral triples on the Sierpiński gasket and other fractal curves

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-)Kaehler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail.

We review the motivation, construction and physical interpretation of a semi-finite spectral triple obtained through a rearrangement of central elements of loop quantum gravity. The triple is based on a countable set of oriented graphs and the algebra consists of generalized holonomy loops in this set. The Dirac-type operator resembles a global functional derivation operator and the interaction between the algebra of holonomy loops and the Dirac-type operator reproduces the structure of a quantized Poisson bracket of general relativity. Finally, we give a heuristic argument as to how a natural candidate for a quantized Hamiltonian might emerge from this spectral triple construction.

We consider a natural class $\mathcal {ULG}$ of connected, simply connected nilpotent Lie groups which contains ℝn, the group $\mathcal {UT}_n(\mathbb {R})$ of all triangular unipotent matrices over ℝ and many of its subgroups, and is closed under direct products. If $G \in \mathcal {ULG}$, then $\Gamma _1 = G\cap \mathcal {UT}_n(\mathbb {Z})$ is a lattice subgroup of G. We prove that if $G \in \mathcal {ULG}$ and Γ is a lattice subgroup of G, then a free ergodic measure-preserving action T of G on a probability space (X,ℬ,μ) has completely positive entropy (CPE) if and only if the restriction TΓ of T to Γ has CPE. We can deduce from this the following version of a well-known conjecture in this case: the action T has CPE if and only if T is uniformly mixing. Moreover, such T has a Lebesgue spectrum with infinite multiplicity. We further consider an ergodic free action T with positive entropy and suppose TΓ is ergodic for any lattice subgroup Γ of G. This holds, in particular, if the spectrum of T does not contain a discrete component. Then we show the Pinsker algebra Π(T) of T exists and coincides with the Pinsker algebras Π(TΓ) of TΓ for any lattice subgroup Γ of G. In this case, T always has Lebesgue spectrum with infinite multiplicity on the space ℒ20(X,μ)⊖ℒ20(Π(T)) , where ℒ20(Π(T)) contains all Π(T) -measurable functions from ℒ20(X,μ) . To prove these results, we use the following formula: h(T)=∣G(Γ)∣−1hK (TΓ) , where h(T) is the Ornstein–Weiss entropy of T, hK (TΓ) is a Kolmogorov–Sinai entropy of TΓ, and the number ∣G(TΓ)∣ is the Haar measure of the compact subset G(Γ) of G. In particular, h(T)=hK (TΓ1) , and hK (TΓ1)=∣G(Γ)∣−1hK (TΓ) . The last relation is an analogue of the Abramov formula for flows.

Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and the functional calculus theorem for bounded self-adjoint operators. Section 1.6 is devoted to Stone’s theorem. In Section 1.7, the polar decomposition of bounded linear operators is extended to closed linear operators. Section 1.8 introduces Nelson’s analytic vector theorem for the selfadjointness of closed symmetric operators. Section 1.9, finally, deals with the form representation theorem and the Friedrichs self-adjoint extension theorem.

In this chapter we shall prove the spectral theorem for self-adjoint (unbounded) operators A in Hilbert spaces. It says that for each A there exists a unique spectral measure E on a(A), so that A = ∫ t dE. In the sequel H is a complex Hilbert space.

We embed the quantum Heisenberg manifold $D_{\mu \nu }^{c}$ in a crossed product $\mathrm {C}^*$-algebra. This enables us to show that all tracial states on $D_{\mu \nu }^{c}$ induce the same homomorphism on $K_0(D_{\mu \nu }^{c})$, whose range is the group $\mathbf {Z} +2\mu \mathbf {Z} + 2\nu \mathbf {Z}$.

Yang-Mills connections on quantum Heisenberg manifolds

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form \({-Au \in \partial\varphi(u)}\) as well ass dissipative initial value evolutions of the form \({-\dot{u}(t) - Au(t) + \omega{u}(t) \in \partial\varphi(l,u(l))}\) where \({\varphi}\) is a convex potential on an infinite dimensional space, A is a linear operator and \({\omega}\) is any scalar. The framework developed in the above mentioned paper reformulates these problems as \({0\in \bar{\partial}L(u)}\) and \({\dot{u}(t) \in \bar{\partial}L(t,u(t))}\) respectively, where \({\bar{\partial}L}\) is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term.

We give a functional analytical definition of the Maslov index for continuous curves in the Fredholm-Lagrangian Grassmannian. Our definition does not require assumptions either at the endpoints or at the crossings of the curve with the Maslov cycle. We demonstrate an application of our definition by developing the symplectic geometry of self-adjoint extensions of unbounded symmetric operators. We discuss continuous variations of the form $A_D+C_t$, where $A_D$ is a fixed self-adjoint unbounded Fredholm operator and $\{C_t\}$ a family of bounded self-adjoint operators. We extend the definition of the spectral flow to such families of unbounded operators in a purely functional analytical way. We then prove that the spectral flow is equal to the Maslov index of the corresponding family of abstract Cauchy data spaces.

In the traditional Hilbert space quantum mechanics, (bounded or unbounded) selfadjoint operators are taken as physical observables. In the present work a more comprehensive approach is used: observables are represented as normalised positive operator measures. We develop the spectral representation theory of generally unbounded selfadjoint operators in this more general setting of positive operator measures, paving the way for important physical applications in the sequel.

In this chapter the perturbation theory developed for bounded Fredholm operators in Sections XI.3 and XI.4 is extended to closed unbounded Fredholm operators. For the additive perturbation theorems this is accomplished via the simple device of renorming the domain of the operator T. In this way T becomes a bounded Fredholm operator to which the theorems of Chapter XI are applicable. We start with some properties and examples of the graph norm. The third and fourth section contain the main perturbation theorems. Section 5 presents a completeness theorem for certain compact perturbations of unbounded selfadjoint operators.