Injective matricial Hilbert spaces

1991 ◽  
Vol 110 (1) ◽  
pp. 183-190 ◽  
Author(s):  
A. Guyan Robertson
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann Algebras ◽  
Linear Span ◽  
Bounded Operator ◽  
Operator Space ◽  
Operator Spaces ◽  
Von Neumann ◽  
Type Iv ◽  
Injective Operator ◽  
Cartan Factor

Injective matricial operator spaces have been classified up to Banach space isomorphism in [20]. The result is that every such space is isomorphic to l∞, l2, B(l2), or a direct sum of such spaces. A more natural project, given the matricial nature of the definitions involved, would be the classification of such spaces up to completely bounded isomorphism. This was done for injective von Neumann algebras in [6] and for injective operator systems (i.e. unital injective operator spaces) in [19]. It turns out that the spaces l∞ and B(l2) are in a natural way uniquely characterized up to completely bounded isomorphism. However, as shown in [20], a problem arises in the case of l2. For there are two injective operator spaces which are each isometrically isomorphic to l2 but not completely boundedly isomorphic to each other. We shall resolve this problem by showing that these are the only two possibilities, in the sense that any injective operator space which is isometric to l2 is completely isometric to one of them. (See Corollary 3 below.) The Hilbert spaces in von Neumann algebras investigated in [17], [13] turn out to be injective matricial operator spaces and are therefore completely isometric to one of our two examples. Another Hilbert space in B(l2) which has been much studied in operator theory, complex analysis and physics is the Cartan factor of type IV [10]. This is the complex linear span of a spin system and generates the Fermion C*-algebra ([3], §5·2). We show that a Cartan factor of type IV is not even completely boundedly isomorphic to an injective matricial operator space. One curious property of all the aforementioned Hilbert spaces is that every bounded operator on them is actually completely bounded, a fact that is crucial in our proofs.

On the Duality of Operator Spaces

1995 ◽  
Vol 38 (3) ◽  
pp. 334-346 ◽  
Author(s):  
Christian Le Merdy
Keyword(s):  
Banach Space ◽  
Von Neumann Algebras ◽  
Related Result ◽  
Space Structure ◽  
Operator Space ◽  
Operator Spaces ◽  
Bounded Operators ◽  
Von Neumann ◽  
Operator Space Structure ◽  
Dual Banach Space

AbstractWe prove that given an operator space structure on a dual Banach space Y*, it is not necessarily the dual one of some operator space structure on Y. This allows us to show that Sakai's theorem providing the identification between C*-algebras having a predual and von Neumann algebras does not extend to the category of operator spaces. We also include a related result about completely bounded operators from B(ℓ2)* into the operator Hilbert space OH.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

scholarly journals Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

2019 ◽  
Vol 31 (04) ◽  
pp. 1950013 ◽  
Author(s):  
Valter Moretti ◽  
Marco Oppio
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Hilbert Spaces ◽  
Von Neumann Algebras ◽  
Complex Structure ◽  
Complex Hilbert Space ◽  
Von Neumann ◽  
The Real ◽  
Quaternionic Hilbert Space ◽  
Adjoint Operators

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

On the Dixmier property of simple C*-algebras

1982 ◽  
Vol 91 (1) ◽  
pp. 75-78 ◽  
Author(s):  
Norbert Riedel
Keyword(s):  
Convex Hull ◽  
Von Neumann Algebras ◽  
Present Note ◽  
Linear Span ◽  
Closed Convex Hull ◽  
Von Neumann ◽  
Tracial State ◽  
C Algebras ◽  
Unital Simple

A unital C*-algebra is said to satisfy the Dixmier property if for each element x in the closed convex hull of all elements of the form u*xu, u being a unitary in , intersects the centre of ((2), 2·7). The von Neumann algebras and also some other classes of C*-algebras are known to satisfy the Dixmier property (cf. (2), (3), (4), (6)). If is a simple C*-algebra which satisfies the Dixmier property then has at most one tracial state. In (3) Archbold raised the question whether there exists a unital simple C*-algebra which has at most one tracial state without satisfying the Dixmier property. In the present note we characterize the unital simple C*-algebras with at most one tracial state in terms of a condition which is similar to the Dixmier property, but is in fact formally weaker in the framework of simple C*-algebras. This characterization relies on the method used by Pedersen in (5) in order to show that for a unital simple C*-algebra which has at most one tracial state and at least one non-trivial projection the linear span of all projections in is dense in As an application we characterize those unital simple C*-algebras with a unique tracial state which satisfy the Dixmier property.

scholarly journals Rigid $\mathcal{OL}_p$structures of non-commutative $L_p$-spaces associated with hyperfinite von Neumann algebras

2005 ◽  
Vol 96 (1) ◽  
pp. 63 ◽  
Author(s):  
Marius Junge ◽  
Zhong-Jin Ruan ◽  
Quanhua Xu
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Space ◽  
Local Operator ◽  
Space Structures ◽  
Von Neumann ◽  
Local Properties

This paper is devoted to the study of rigid local operator space structures on non-commutative $L_p$-spaces. We show that for $1\le p \neq 2 < \infty$, a non-commutative $L_p$-space $L_p(\mathcal M)$ is a rigid $\mathcal{OL}_p$ space (equivalently, a rigid $\mathcal{COL}_p$ space) if and only if it is a matrix orderly rigid $\mathcal{OL}_p$ space (equivalently, a matrix orderly rigid $\mathcal{COL}_p$ space). We also show that $L_p(\mathcal M)$ has these local properties if and only if the associated von Neumann algebra $\mathcal M$ is hyperfinite. Therefore, these local operator space properties on non-commutative $L_p$-spaces characterize hyperfinite von Neumann algebras.

Type Decomposition and the Rectangular AFD Property for W*-TRO’s

2004 ◽  
Vol 56 (4) ◽  
pp. 843-870 ◽  
Author(s):  
Zhong-Jin Ruan
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Space ◽  
Dual Operator ◽  
Von Neumann ◽  
Direct Sum ◽  
Cardinal Numbers ◽  
Finite Dimensional ◽  
Dimensional Property ◽  
Type Decomposition

AbstractWe study the type decomposition and the rectangular AFD property for W*-TRO’s. Like von Neumann algebras, every W*-TRO can be uniquely decomposed into the direct sum of W*- TRO's of type I, type II, and type III. We may further considerW*-TRO's of type Im,n with cardinal numbers m and n, and considerW*-TRO's of type IIλ,μ with λ, μ = 1 or ∞. It is shown that every separable stable W*-TRO (which includes type I∞, ∞, type II∞, ∞ and type III) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for W*-TRO’s. One of our major results is to show that a separable W*-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular space (equivalently, a rectangular space).

scholarly journals The “maximal” tensor product of operator spaces

1999 ◽  
Vol 42 (2) ◽  
pp. 267-284 ◽  
Author(s):  
Timur Oikhberg ◽  
Gilles Pisier
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
Tensor Products ◽  
Operator Space ◽  
Operator Spaces ◽  
Two Factors

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

scholarly journals Multiplicity-freeness of Unitary Representations in Sections of Holomorphic Hilbert Bundles

2018 ◽  
Vol 2020 (16) ◽  
pp. 4852-4889
Author(s):  
Martín Miglioli ◽  
Karl-Hermann Neeb
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann Algebras ◽  
Unitary Representations ◽  
Banach Manifold ◽  
Von Neumann ◽  
Holomorphic Bundle ◽  
Holomorphic Sections ◽  
Banach Lie Group ◽  
Visible Action ◽  
Multiplicity Free

Abstract We prove several results asserting that the action of a Banach–Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity-free. These results require the existence of compatible anti-holomorphic bundle maps and certain multiplicity-freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,\sigma )$-weakly visible action (generalizing T. Koboyashi’s visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for group actions on homogeneous bundles for larger groups. We illustrate these general results by several examples related to operator groups and von Neumann algebras.

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