Underlying Von Neumann Algebra Research Articles

Nonassociative Lp-spaces and embeddings in noncommutative Lp-spaces

We define a notion of nonassociative Lp-space associated to a JBW⁎-algebra (Jordan von Neumann algebra) equipped with a normal faithful state φ. In the particular case of JW⁎-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative Lp-spaces. We also make the link with the nonassociative Lp-spaces of Iochum associated to JBW-algebras and the investigation of contractively complemented subspaces of noncommutative Lp-spaces. More precisely, we show that our nonassociative Lp-spaces contain isometrically the Lp-spaces of Iochum and that all tracial nonassociative Lp-spaces from JW⁎-factors arise as positively contractively complemented subspaces of noncommutative Lp-spaces.

Maps between positive cones of operator algebras preserving a measure of the difference between arithmetic and geometric means

On the set of positive invertible elements in a finite von Neumann algebra carrying a faithful normalized trace $$\tau $$ the numerical quantity $$\begin{aligned} d_{\tau }(A,B)=\tau (A + B)/2 - \tau \left( A^{1/2}\left( A^{-1/2}BA^{-1/2}\right) ^{1/2}A^{1/2}\right) \end{aligned}$$can be viewed as a measure of the difference of the arithmetic and the geometric mean. In this paper, we study maps between the positive definite cones of operator algebras which respect the above distance measure. We obtain the interesting fact that any such map originates from a trace-preserving Jordan $${}^*$$-isomorphisms (either algebra $${}^*$$-isomorphism or algebra $${}^*$$-antiisomorphism in the more restrictive case of factors) between the underlying von Neumann algebras.

The positive contractive part of a noncommutative Lp-space is a complete Jordan invariant

Let 1≤p≤+∞. We show that the positive part of the closed unit ball of a non-commutative Lp-space, as a metric space, is a complete Jordan ⁎-invariant for the underlying von Neumann algebra.

Twisting non-commutative Lp spaces

The paper makes the first steps into the study of extensions (“twisted sums”) of noncommutative Lp-spaces regarded as Banach modules over the underlying von Neumann algebra M. Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss, saying that whenever one finds a given Banach space Y as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension 0⟶Y⟶X⟶Y⟶0.For semifinite algebras, considering Lp=Lp(M,τ) as an interpolation space between M and its predual M⁎ one arrives at a certain self-extension of Lp that is a kind of noncommutative Kalton–Peck space and carries a natural bimodule structure. Some interesting properties of these spaces are presented.For general algebras, including those of type III, the interpolation mechanism produces two (rather than one) extensions of one sided modules, one of left-modules and the other of right-modules. Whether or not one may find (nontrivial) self-extensions of bimodules in all cases is left open.

Transition probabilities of normal states determine the Jordan structure of a quantum system

Let Φ : 𝔖(M1) → 𝔖(M2) be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras M1 and M2, respectively. This paper concerns with the situation when Φ preserves (or partially preserves) one of the following three notions of “transition probability” on the normal state spaces: the transition probability PU introduced by Uhlmann [Rep. Math. Phys. 9, 273-279 (1976)], the transition probability PR introduced by Raggio [Lett. Math. Phys. 6, 233-236 (1982)], and an “asymmetric transition probability” P0 (as introduced in this article). It is shown that the two systems are isomorphic, i.e., M1 and M2 are Jordan ∗-isomorphic, if Φ preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states μ and ν, we have PΦ(μ),Φ(ν)=0 if and only if P(μ, ν) = 0, where P stands for PU (respectively, PR or P0). Furthermore, as an extension of Wigner’s theorem, it is shown that there is a Jordan ∗-isomorphism Θ : M2 → M1 satisfying Φ = Θ∗|𝔖(M1) if and only if Φ preserves the “asymmetric transition probability.” This is also equivalent to Φ preserving the Raggio transition probability. Consequently, if Φ preserves the Raggio transition probability, it will preserve the Uhlmann transition probability as well. As another application, the sets of normal states equipped with either the usual metric, the Bures metric or “the metric induced by the self-dual cone,” are complete Jordan ∗-invariants for the underlying von Neumann algebras.

Multiplicative Jordan triple isomorphisms on the self-adjoint elements of von Neumann algebras

In this paper we consider multiplicative Jordan triple isomorphisms between the sets of self-adjoint elements (respectively the sets of positive elements) of von Neumann algebras. These transformations are the bijective maps which satisfy the equality ϕ ( ABA ) = ϕ ( A ) ϕ ( B ) ϕ ( A ) on their domains. We show that all those transformations originate from linear ∗-algebra isomorphisms and linear ∗-algebra antiisomorphisms in the case when the underlying von Neumann algebras do not have commutative direct summands. An application of our results concerning non-linear maps which preserve the absolute value of products is also presented.

Embedding of Cq and Rq into noncommutative Lp -spaces, 1≤p<q≤2

We prove that a quotient of a subspace of C p ⊕ p R p (1≤p<2) embeds completely isomorphically into a noncommutative L p -space, where C p and R p are respectively the p-column and p-row Hilbertian operator spaces. We also represent C q and R q (p<q≤2) as quotients of subspaces of C p ⊕ p R p . Consequently, C q and R q embed completely isomorphically into a noncommutative L p (M). We further show that the underlying von Neumann algebra M cannot be semifinite.

Some physical appearances of vector coherent states and coherent states related to degenerate Hamiltonians

In the spirit of some earlier work on the construction of vector coherent states (VCS) over matrix domains, we compute here such states associated to some physical Hamiltonians. In particular, we construct vector coherent states of the Gazeau–Klauder type. As a related problem, we also suggest a way to handle degeneracies in the Hamiltonian for building coherent states. Specific physical Hamiltonians studied include a single photon mode interacting with a pair of fermions, a Hamiltonian involving a single boson and a single fermion, a charged particle in a three-dimensional harmonic force field and the case of a two-dimensional electron placed in a constant magnetic field, orthogonal to the plane which contains the electron. In this last example, which is related to the fractional quantum Hall effect, an interesting modular structure emerges for two underlying von Neumann algebras, related to opposite directions of the magnetic field. This leads to the existence of coherent states built out of Kubo-Martin-Schwinger (KMS) states for the system.

Formula omitted] spaces—the local structure of non-commutative Lp spaces

Developing the theory of COL p spaces (a variation of the non-commutative analogue of L p spaces), we provide new tools to investigate the local structure of non-commutative L p spaces. Under mild assumptions on the underlying von Neumann algebras, non-commutative L p spaces with Grothendieck's approximation property behave locally like the space of matrices equipped with the p-norm (of the sequences of their singular values). As applications, we obtain a basis for non-commutative L p spaces associated with hyperfinite von Neumann algebras with separable predual von Neumann algebras generated by free groups, and obtain a basis for separable nuclear C ∗ -algebras.

A Locally Compact Quantum Group Analogue of the Normalizer of SU(1,1) in SL(2, ℂ)

S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently by Woronowicz gave strong indications that the normalizer N of SU(1,1) in SL(2,C) is a much better quantization candidate than SU(1,1) itself. In this paper we show that this is indeed the case by constructing N_q, a new example of a unimodular locally compact quantum group (depending on a parameter q) that is a deformation of N. After defining the underlying von Neumann algebra of N_q we use a certain class of q-hypergeometric functions and their orthogonality relations to construct the comultiplication. The coassociativity of this comultiplication is the hardest result to establish. We define the Haar weight and obtain simple formulas for the antipode and its polar decomposition. As a final result we produce the underlying C*-algebra of N_q. The proofs of all these results depend on various properties of q-hypergeometric 1\phi1 functions.

Non-Commutative Bounded Vilenkin Systems

We consider orthonormal systems in spaces of measurable operators associated with a finite von Neumann algebra which contain the classical bounded Vilenkin systems. We show that they form Schauder bases in all reflexive non-commutative Lp-spaces when taken in the lexicographic order. This is a non-commutative analogue of a theorem of Paley. The principal theme of this paper finds its source in the classical theorems of Paley [Pa]. In the terminology of the theory of Banach spaces, these theorems assert that the classical Walsh system, taken in the Walsh-Paley ordering (respectively, partitioned into dyadic blocks) forms a (Schauder) basis (respectively, an unconditional decomposition) in each of the spaces Lpâ0;1a; 1 < p <1. The classical Walsh system may be identified with the dual group of the familiar dyadic group. From this point of view, the notion of a Vilenkin system introduced in [Vi] generalizes that of the Walsh system. Watari [Wa] extended Paley’s theorems to bounded Vilenkin systems and showed that such systems when enumerated lexicographically not only form a Schauder basis in each of the reflexive Lp-spaces, but also permit a finer unconditional decomposition than that given by the dyadic decomposition associated with the Walsh system. The study of orthogonal series in the setting of non-commutative Lpspaces was initiated in [SF1,2]. In this setting, the classical Walsh system is replaced by a system of eigenoperators arising from the action of a compact Abelian group on the underlying von Neumann algebra. The methods of [SF1,2] are based on the fact that non-commutative Lp-spaces have the unconditionality property for martingale difference sequences and this approach permits an extension of Paley’s theorems to the non-commutative setting. Our intention in this paper is to study orthogonal systems in spaces of measurable operators affiliated with a finite von Neumann algebra, which

Physical Symmetries of Quantum Histories

Gell-Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals play a key role. Physical symmetries of quantum history systems are shown, under mild physical assumptions, to correspond to isometric Jordan *-automorphisms of the underlying von Neumann algebra. Unitary representations of groups of symmetries of quadratic forms over operator algebras are discussed and related to symmetries of decoherence functionals.

An Application of Orthoisomorphisms to Non-Commutative $L^p$-Isometries

We prove that if there exists an into linear isometry between non-commutative Lp-spaces then there exists an into Jordan \* -isomorphism between underlying von Neumann algebras, as an application of Araki-Bunce-Wright's theorem concerning the characterization of orthogonality preserving positive maps between preduals. Moreover, we determine the structure of a linear non-commutative Lp-isometry when it is surjective and \*-preserving.