GNS Representation Research Articles

Geometric representation theory for unitary groups of operator algebras

Geometric realizations for the restrictions of GNS representations to unitary groups of C ∗ -algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach–Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.

Background independent quantizations—the scalar field: I

We are concerned with the issue of quantization of a scalar field in a diffeomorphism-invariant manner. We apply the method used in loop quantum gravity. It relies on a specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the ‘quantum’ polymer *-star algebra and looking for positive linear functionals, called states. It is assumed in this paper that homeomorphism invariance allows us to determine a complete class of the states. Except one, all of them are new. In this paper we outline the main steps and conclusions, and present the results: the GNS representations, characterization of those states which lead to essentially self-adjoint momentum operators (unbounded), identification of the equivalence classes of the representations as well as of the irreducible ones.

Bose-Einstein condensate and spontaneous breaking of conformal symmetry on Killing horizons II

In a previous paper (hep-th/0407256) local scalar QFT (in Weyl algebraic approach) has been constructed on degenerate semi-Riemannian manifolds $\bS^1\times \Sigma$ corresponding to the extension of Killing horizons by adding points at infinity to the null geodesic forming the horizon. It has been proved that the theory admits a natural representation of $PSL(2,\bR)$ in terms of $*$-automorphisms and this representation is unitarily implementable if referring to a certain invariant state $\lambda$. Among other results it has been proved that the theory admits a class of inequivalent algebraic (coherent) states $\{\lambda_\zeta\}$, with $\zeta\in L^2(\Sigma)$, which break part of the symmetry, in the sense that each of them is not invariant under the full group $PSL(2,\bR)$ and so there is no unitary representation of whole group $PSL(2,\bR)$ which leaves fixed the cyclic GNS vector. These states, if restricted to suitable portions of $\bM$ are invariant and extremal KMS states with respect a surviving one-parameter group symmetry. In this paper we clarify the nature of symmetry breakdown. We show that, in fact, {\em spontaneous} symmetry breaking occurs in the natural sense of algebraic quantum field theory: if $\zeta \neq 0$, there is no unitary representation of whole group $PSL(2,\bR)$ which implements the $*$-automorphism representation of $PSL(2,\bR)$ itself in the GNS representation of $\lambda_\zeta$ (leaving fixed or not the state).

Bose–Einstein condensate and spontaneous breaking of conformal symmetry on Killing horizons

Local scalar quantum field theory (in Weyl algebraic approach) is constructed on degenerate semi-Riemannian manifolds corresponding to Killing horizons in spacetime. Covariance properties of the C*-algebra of observables with respect to the conformal group PSL(2,R) are studied. It is shown that, in addition to the state studied by Guido, Longo, Roberts, and Verch for bifurcated Killing horizons, which is conformally invariant and KMS at Hawking temperature with respect to the Killing flow and defines a conformal net of von Neumann algebras, there is a further wide class of algebraic (coherent) states representing spontaneous breaking of PSL(2,R) symmetry. This class is labeled by functions in a suitable Hilbert space and their GNS representations enjoy remarkable properties. The states are nonequivalent extremal KMS states at Hawking temperature with respect to the residual one-parameter subgroup of PSL(2,R) associated with the Killing flow. The KMS property is valid for the two local subalgebras of observables uniquely determined by covariance and invariance under the residual symmetry unitarily represented. These algebras rely on the physical region of the manifold corresponding to a Killing horizon cleaned up by removing the unphysical points at infinity [necessary to describe the whole PSL(2,R) action]. Each of the found states can be interpreted as a different thermodynamic phase, containing Bose–Einstein condensate, for the considered quantum field. It is finally suggested that the found states could describe different black holes.

Some Properties of a Class of Diagonalizable States of von Neumann Algebras

In this paper, a class of representations of uniformly hyperfinite algebras is constructed and the corresponding von Neumann algebras are studied. It is proved that, under certain conditions, the Markov states generate factors of type IIIλ, where λ ∈ (0,1), in the GNS representation; this gives a negative answer to the conjecture that the factors corresponding to Hamiltonians with nontrivial interactions have type III. It is shown that, for a certain class of Hamiltonians, there exists a unique translation-invariant ground state.

Relative Dixmier property and complete boundedness of modular maps

In this paper, we are concerned with the connection between the relative Dixmier property and the complete boundedness of bounded modular maps. For an inclusion of II1-factors with the relative Dixmier property, we will show that is properly infinite where π is the GNS representation associated with a singular state on . We show that a bounded -multimodular bilinear map is completely bounded and that a bounded -bimodule map is completely bounded. Finally, we prove the row boundedness of a normal -bimodule map where is hyperfinite without the relative Dixmier property.

The Martin boundary of a discrete quantum group

We consider the Markov operator Pφ on a discrete quantum group given by convolution with a q-tracial state φ. In the study of harmonic elements x, Pφ(x) = x, we define the Martin boundary Aφ. It is a separable C ∗-algebra carrying canonical actions of the quantum group and its dual. We establish a representation theorem to the effect that positive harmonic elements correspond to positive linear functionals on Aφ. The C ∗-algebraAφ has a natural time evolution, and the unit can always be represented by a KMS state. Any such state gives rise to a u.c.p. map from the von Neumann closure of Aφ in its GNS representation to the von Neumann algebra of bounded harmonic elements, which is an analogue of the Poisson integral. Under additional assumptions this map is an isomorphism which respects the actions of the quantum group and its dual. Next we apply these results to identify the Martin boundary of the dual of SUq(2) with the quantum homogeneous sphere of Podleś. This result extends and unifies previous results by Ph. Biane and M. Izumi.

On Energy-Momentum Spectrum of Stationary States with Nonvanishing Current on 1-d Lattice Systems

On one-dimensional two-way infinite quantum lattice system, a property of translationally invariant stationary states with nonvanishing current expectation is investigated. We consider GNS representation with respect to such a state, on which we have a group of space-time translation unitary operators. We show that spectrum of the unitary operators, energy-momentum spectrum with respect to the state, has a singularity at the origin.

Polarized States on the Weyl Algebra

A state σ on the Weyl algebra A(V, Ω) over a real symplectic vector space (V, Ω) is polarized when the vectors v for which the absolute value of σ on the corresponding Weyl generator δ_v is unity constitute a maximal Ω -integral additive subgroup of V ; such a state is necessarily pure, but has inseparable carrier space and is not regular. We determine fundamental properties of such states: in particular, we decide precisely when the GNS representations associated to a pair of polarized states are unitarily equivalent and decide precisely when a given symplectic automorphism is unitarily implemented in the GNS representation associated to a given polarized state.

Dirichlet forms and symmetric Markovian semigroups on CCR algebras with respect to quasi-free states

Employing the construction method of Dirichlet forms on standard forms of von Neumann algebras developed in Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2000, Vol. 3, No. 1, pp. 1–14 (Ref. 1), we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let A(h0) be the CCR algebra over a complex separable pre-Hilbert space h0 and let ω be a quasi-free state on A(h0). For any normalized admissible function f and complete orthonormal system (CONS) {gn}⊂h0, we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of (A(h0),ω). It turns out that the form is independent of admissible function f and CONS {gn} chosen. By analyzing the spectrum of the generator (Dirichlet operator) of the semigroup, we show that the semigroup is ergodic and tends to the equilibrium exponentially fast.

A UNIFIED APPROACH TO GENERALIZED CONDITIONAL EXPECTATIONS OPERATOR VALUED WEIGHTS AND RADON–NIKODYM DERIVATIVES ON VON NEUMANN ALGEBRAS

Given two von Neumann algebras, ℳ and [Formula: see text] with [Formula: see text], and two normal semifinite faithful weights, φ and ψ on ℳ and [Formula: see text] respectively, we define a canonical map from {b ∈ ℳ+ | φ(b)< ∞} to the set of positive forms on the Hilbert space of the GNS representation of [Formula: see text] associated to ψ. We show that generalized conditional expectations, operator valued weights and Radon–Nikodym derivatives on von Neumann algebras can be obtained from particular cases of this canonical map.

A remark on the deformation of GNS representations of *-algebras

Motivated by deformation quantization we investigate an algebraic GNS construction of *-representations of deformed *-algebras over ordered rings and compute their classical limit. The question whether a GNS representation can be deformed leads to deformation of positive linear functionals. Various physical examples from deformation quantization like the Bargmann—Fock and the Schrödinger representation as well as KMS functionals are discussed.

Filtered random variables, bialgebras, and convolutions

We introduce the filtered *-bialgebra which is a multivariate generalization of the unital *-bialgebra C〈X,X′,P〉 of polynomials in noncommuting variables X=X*, X′*=X′ and a projection P=P*=P2, endowed with the coproduct Δ(X)=X⊗1+1⊗X, Δ(X′)=X′⊗P+P⊗X′, with P being group-like. We study the associated convolutions, random walks and filtered random variables. The GNS representations of the limit states lead to filtered fundamental operators which are the CCR fundamental operators on the multiple symmetric Fock space Γ(ℋ) over H=L2(R+,G), where 𝒢 is a separable Hilbert space, multiplied by appropriate projections. The importance of filtered random variables and fundamental operators stems from the fact that by addition and strong limits one obtains from them the main types of noncommutative random variables and fundamental operators, respectively, regardless of the type of noncommutative independence.

A quantum spin system with random interactions I

We study a quantum spin glass as a quantum spin system with random interactions and establish the existence of a family of evolution groups {τt(ω)}ω∈/Ω of the spin system. The notion of ergodicity of a measure preserving group of automorphisms of the probability space Ω, is used to prove the almost sure independence of the Arveson spectrum Sp(τ(ω)) of τt(e). As a consequence, for any family of (τ(ω),β) — KMS states {ρ(ω)}, the spectrum of the generator of the group of unitaries which implement τ(ω) in the GNS representation is also almost surely independent of ω.

Invariant Hermitian kernels and their Kolmogorov decompositions

We study Hermitian kernels invariant under the action of a semigroup with involution. We characterize the Hermitian kernels that realize the given action by bounded operators on a Kreı̆n space. Applications include the GNS representation of ∗-algebras associated to Hermitian functionals and the dilation theory of Hermitian maps on C ∗ -algebras.

Weak Hopf Algebras: II. Representation Theory, Dimensions, and the Markov Trace

If A is a weak C*-Hopf algebra then the category of finite-dimensional unitary representations of A is a monoidal C*-category with its monoidal unit being the GNS representation Dε associated to the counit ε. This category has isomorphic left dual and right dual objects, which leads, as usual, to the notion of a dimension function. However, if ε is not pure the dimension function is matrix valued with rows and columns labeled by the irreducibles contained in Dε. This happens precisely when the inclusions AL⊂A and AR⊂A are not connected. Still, there exists a trace on A which is the Markov trace for both inclusions. We derive two numerical invariants for each C*-WHA of trivial hypercenter. These are the common indices I and δ, of the Haar, respectively Markov, conditional expectations of either one of the inclusions AL/R⊂A or ÁL/R⊂Á. In generic cases I>δ. In the special case of weak Kac algebras we reproduce D. Nikshych's result (2000, J. Operator Theory, to appear) by showing that I=δ and is always an integer.

Representations of Hermitian Kernels¶by Means of Krein Spaces.¶II. Invariant Kernels

In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kreîn space. This is motivated by the GNS representation of *-algebras associated to hermitian functionals, the dilation theory of hermitian maps on C *-algebras, as well as others. We explain the key role played by the technique of induced Kreîn spaces and a lifting property associated to them.

Locality in GNS Representations of Deformation Quantization

In the framework of deformation quantization we apply the formal GNS construction to find representations of the deformed algebras in pre-Hilbert spaces over $\mathbb C[[\lambda]]$ and establish the notion of local operators in these pre-Hilbert spaces. The commutant within the local operators is used to distinguish `thermal' from `pure' representations. The computation of the local commutant is exemplified in various situations leading to the physically reasonable distinction between thermal representations and pure ones. Moreover, an analogue of von Neumann's double commutant theorem is proved in the particular situation of a GNS representation with respect to a KMS functional and for the Schr\"odinger representation on cotangent bundles. Finally we prove a formal version of the Tomita-Takesaki theorem.

Homogeneous Fedosov star products on cotangent bundles II: GNS representations, the WKB expansion, traces, and applications

This paper is part II of a series of papers on the deformation quantization on the cotangent bundle of an arbitrary manifold Q. For certain homogeneous star products of Weyl ordered type (which we have obtained from a Fedosov type procedure in part I, see [M. Bordemann, N. Neumaier, S. Waldmann, Homogeneous Fedosov star products on cotangent bundles I: Weyl and standard ordering with differential operator representation, Comm. Math. Phys. 198 (1998) 363–396]) we construct differential operator representations via the formal GNS construction (see [M. Bordemann, S. Waldmann, Formal GNS construction and states in deformation quantization, Comm. Math. Phys. 195 (1998) 549–583]). The positive linear functional is integration over Q with respect to some fixed density and is shown to yield a reasonable version of the Schrödinger representation where a Weyl ordering prescription is incorporated. Furthermore we discuss simple examples like free particle Hamiltonians (defined by a Riemannian metric on Q) and the implementation of certain diffeomorphisms of Q to unitary transformations in the GNS (pre-)Hilbert space and of time reversal maps (involutive anti-symplectic diffeomorphisms of T ∗ Q ) to anti-unitary transformations. We show that the fixed-point set of any involutive time reversal map is either empty or a Lagrangian submanifold. Moreover, we compare our approach to concepts using integral formulas of generalized Moyal-Weyl type. Furthermore we show that the usual WKB expansion with respect to a projectable Lagrangian submanifold can be formulated by a GNS construction. Finally we prove that any homogeneous star product on any cotangent bundle is strongly closed, i.e. the integral over T ∗ Q w.r.t. the symplectic volume vanishes on star-commutators. An alternative Fedosov type deduction of the star product of standard ordered type using a deformation of the algebra of symmetric contravariant tensor fields is given.

Graded KMS-functionals and the breakdown of supersymmetry

It is shown that the modulus of any graded or, more generally, twisted KMS– functional of a C–dynamical system is proportional to an ordinary KMS–state and the twist is weakly inner in the corresponding GNS–representation. If the functional is invariant under the adjoint action of some asymptotically abelian family of automorphisms, then the twist is trivial. As a consequence, such functionals do not exist for supersymmetric C–dynamical systems. This is in contrast with the situation in compact spaces where super KMS–functionals occur as super-Gibbs functionals. ∗Supported in part by GNAFA and MURST.