Commutativity Preserving Maps of Factors

Sturm–Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Prüfer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

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.

Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

In this paper, we propose a new modification of the Gradient Projection Algorithm and the Forward–Backward Algorithm. Using our proposed algorithms, we establish two strong convergence theorems for solving convex minimization problem, monotone variational inclusion problem and fixed point problem for demicontractive mappings in a real Hilbert space. Furthermore, we apply our results to solve split feasibility and optimal control problems. We also give two numerical examples of our algorithm in real Euclidean space of dimension 4 and in an infinite dimensional Hilbert space, to show the efficiency and advantage of our results.

Let H be an infinite dimensional complex Hilbert space and let be a surjective linear map on B(H) with (I) I 2 K(H), where K(H) denotes the closed ideal of all compact operators on H. If preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then is an automorphism of the algebra B(H). Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

Sequences of [formula omitted]-term deviations in Hilbert space

The purpose of this paper is to compute the filtering equations when the state process, given as the solution of a stochastic differential equation on an infinite dimensional Hilbert space, is observed through a counting observation.

The purpose of this paper is to compute the risk-sensitive filtering equations when the state process, given as the solution of a stochastic differential equation on an infinite dimensional Hilbert space, is observed through a counting observation.

Nonnegative Moore–Penrose inverses of Gram operators

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.

The Banach-Stone theorem says, in the language we shall use, that if two commutative C*-algebras are isomorphic, then they are as C*-algebras. In this formulation there are two natural question: whether the theorem holds for noncommutative C*-algebras, and whether it is possible to replace isometrically isomorphic with isomorphic. A complete answer to the first question was given by Kadison in [10], and M. Cambern settled the second in the affirmative in [5]. In Section 3 we show that Cambern's result may be partially extended to noncommutative von Neumann algebras having the property P ([9, 18, 19]). When we have the result of Section 3 that a nearly isometric completely positive map of a von Neumann algebra having property P is close to a * isomorphism between the algebras, we have nearly proved that von Neumann algebras whose unit balls are close in the Hausdorff metric are unitarily equivalent by a unitary close to the identity. The only thing we have to prove then is that if q) is an isomorphism of a v o n Neumann algebra A having property P onto a v o n Neumann algebra B acting on the same Hilbert space, and 4~ satisfies, for some positive k less than one and any x in A,

In this paper we consider an augmented Lagrangian method for the minimization of a nonlinear functional in the presence of an equality constraint whose image space is in a Hilbert space, an inequality constraint whose image space is finite dimensional, and an affine inequality constraint whose image space is in an infinite dimensional Hilbert space. We obtain local convergence of this method without imposing strict complementarity conditions when the equality, as well as the inequality constraint with finite dimensional image space are augmented. To the author's knowledge this result even generalizes the convergence results which are known when all spaces are finite dimensional.

Let E, F be exact operator spaces (for example subspaces of the C*-algebra K(H) of all the compact operators on an infinite dimensional Hilbert space H). We study a class of bounded linear maps u: E → F* which we call tracially bounded. In particular, we prove that every completely bounded (in short c.b.) map u: E → F* factors boundedly through a Hilbert space. This is used to show that the set OS n of all n-dimensional operator spaces equipped with the c.b. version of the Banach Mazur distance is not separable if n > 2.

Let E, F be exact operator spaces (for example subspaces of the C*-algebra K(H) of all the compact operators on an infinite dimensional Hilbert space H). We study a class of bounded linear maps u: E → F* which we call tracially bounded. In particular, we prove that every completely bounded (in short c.b.) map u: E → F* factors boundedly through a Hilbert space. This is used to show that the set OS n of all n-dimensional operator spaces equipped with the c.b. version of the Banach Mazur distance is not separable if n > 2.As an application we show that there is more than one C*-norm on B(H) ⊗ B(H), or equivalently that $$B(H) \otimes _{\min } B(H) \ne B(H) \otimes _{\max } B(H),$$ which answers a long standing open question.Finally we show that every “maximal” operator space (in the sense of Blecher-Paulsen) is not exact in the infinite dimensional case, and in the finite dimensional case, we give a lower bound for the “exactness constant”. In the final section, we introduce and study a new tensor product for C*-albegras and for operator spaces, closely related to the preceding results.KeywordsHilbert SpaceTensor ProductOperator SpaceBilinear FormFinite Dimensional SubspaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.