Predual Of Von Neumann Algebra Research Articles

Octahedral norms in free Banach lattices

In this paper, we study octahedral norms in free Banach lattices FBL[E] generated by a Banach space E. We prove that if E is an $$L_1(\mu )$$ -space, a predual of von Neumann algebra, a predual of a JBW $$^*$$ -triple, the dual of an M-embedded Banach space, the disc algebra or the projective tensor product under some hypothesis, then the norm of FBL[E] is octahedral. We get the analogous result when the topological dual $$E^*$$ of E is almost square. We finish the paper by proving that the norm of the free Banach lattice generated by a Banach space of dimension $$ \ge 2$$ is nowhere Frechet differentiable. Moreover, we discuss some open problems on this topic.

Continuous model theories for von Neumann algebras

We axiomatize in continuous logic for metric structures σ-finite W⁎-probability spaces and preduals of von Neumann algebras jointly with a weak-* dense C⁎-algebra of its dual. This corresponds respectively to the Ocneanu ultrapower and the Groh ultrapower of (σ-finite in the first case) von Neumann algebras. We give various axiomatizability results corresponding to recent results of Ando and Haagerup including axiomatizability of IIIλ factors for 0<λ≤1 fixed and their preduals. We also strengthen the concrete Groh theory to an axiomatization result for preduals of von Neumann algebras in the language of tracial matrix-ordered operator spaces, a natural language for preduals of dual operator systems. We give an application to the isomorphism of ultrapowers of factors of type III and II∞ for different ultrafilters.

Measures of weak non-compactness in preduals of von Neumann algebras and JBW⁎-triples

We prove, among other results, that three standard measures of weak non-compactness coincide in preduals of JBW⁎-triples. This result is new even for preduals of von Neumann algebras. We further provide a characterization of JBW⁎-triples with strongly WCG predual and describe the order of seminorms defining the strong⁎ topology. As a byproduct we improve a characterization of weakly compact subsets of a JBW⁎-triple predual, providing so a proof for a conjecture, open for almost eighteen years, on weakly compact operators from a JB⁎-triple into a complex Banach space.

LEFT SYMMETRIC POINTS FOR BIRKHOFF ORTHOGONALITY IN THE PREDUALS OF VON NEUMANN ALGEBRAS

In this paper, we give a complete description of left symmetric points for Birkhoff orthogonality in the preduals of von Neumann algebras. As a consequence, except for $\mathbb{C}$, $\ell _{\infty }^{2}$ and $M_{2}(\mathbb{C})$, there are no von Neumann algebras whose preduals have nonzero left symmetric points for Birkhoff orthogonality.

Tingley's problem through the facial structure of operator algebras

Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a real linear surjective isometry between the whole spaces. In this paper, we give an affirmative answer to Tingley's problem when both spaces are preduals of von Neumann algebras, the spaces of self-adjoint operators in von Neumann algebras or the spaces of self-adjoint normal functionals on von Neumann algebras. We also show that every surjective isometry between the unit spheres of unital C⁎-algebras restricts to a bijection between their unitary groups. In addition, we show that every surjective isometry between the normal state spaces or the normal quasi-state spaces of two von Neumann algebras extends to a linear surjective isometry.

A survey on Tingley’s problem for operator algebras

We survey the most recent results on extension of isometries between special subsets of the unit spheres of C$^*$-algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras, and $p$-Schatten-von Neumann spaces, with special interest on Tingley's problem.

Weak* fixed point property of reduced Fourier–Stieltjes algebra and generalization of Baggett's theorem

In this paper, we show that if the reduced Fourier–Stieltjes algebra Bρ(G) of a second countable locally compact group G has either weak* fixed point property or asymptotic center property, then G is compact. As a result, we give affirmative answers to open problems raised by Fendler and et al. in 2013. We then conclude that a second countable group with a discrete reduced dual must be compact. This generalizes a theorem of Baggett. We also construct a compact scattered Hausdorff space Ω for which the dual of the scattered C*-algebra C(Ω) lacks weak* fixed point property. The C*-algebra C(Ω) provides a negative answer to a question of Randrianantoanina in 2010. In addition, we prove a variant of Bruck's generalized fixed point theorem for the preduals of von Neumann algebras. Furthermore, we give some examples emphasizing that the conditions in Bruck's generalized conjecture (BGC) can not be weakened any more.

Decompositions of preduals of JBW and JBW⁎ algebras

We prove that the predual of any JBW⁎-algebra is a complex 1-Plichko space and the predual of any JBW-algebra is a real 1-Plichko space. I.e., any such space has a countably 1-norming Markushevich basis, or, equivalently, a commutative 1-projectional skeleton. This extends recent results of the authors who proved the same for preduals of von Neumann algebras and their self-adjoint parts. However, the more general setting of Jordan algebras turned to be much more complicated. We use in the proof a set-theoretical method of elementary submodels. As a byproduct we obtain a result on amalgamation of projectional skeletons.

On Markushevich bases in preduals of von Neumann algebras

We prove that the predual of any von Neumann algebra is $1$-Plichko, i.e., it has a countably $1$-norming Markushevich basis. This answers a question of the third author who proved the same for preduals of semifinite von Neumann algebras. As a corollary we obtain an easier proof of a result of U.~Haagerup that the predual of any von Neumann algebra enjoys the separable complementation property. We further prove that the self-adjoint part of the predual is $1$-Plichko as well.

$$L_1$$ L 1 -space for a positive operator affiliated with von Neumann algebra

In this paper we suggest an approach for constructing an L1-type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we intro- duce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an L1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent L1- type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that L1 -type space is isometrically isomorphic to the predual of von Neumann algebra in a natural way. We give a small list of alternate definitions of the seminorm, and a special definition for the case of semifinite von Neumann algebra, in particular. We study order properties of L1-type space, and demonstrate the con- nection between semifinite normal weights and positive elements of this space. At last, we construct a similar L-space for the positive element of C*-algebra, and study the connection between this L-space and the L1 -type space in case when this C*-algebra is a von Neumann algebra.

Ideals of operators onC∗-algebras and their preduals

In this paper we consider we study various classical operator ideals (for instance, the ideals of strictly (co)singular, weakly compact, Dunford- Pettis operators) either on C � -algebras, or preduals of von Neumann algebras.

Metrically projective quantum normed spaces that are preduals of von Neumann algebras

Metrically projective quantum normed spaces which are predual for von Neumann algebras are completely characterized.

On the geometry of von Neumann algebra preduals

Let $M$ be a von Neumann algebra and let $M_\star$ be its (unique) predual. We study when for every $\varphi\in M_\star$ there exists $\psi\in M_\star$ solving the equation $\|\varphi \pm \psi\|=\|\varphi\|=\|\psi\|$. This is the case when $M$ does not contain type I nor type III$_1$ factors as direct summands and it is false at least for the unique hyperfinite type III$_1$ factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of $M_\star$ of length 4. An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.

On Perturbed Substochastic Semigroups in Abstract State Spaces

The object of this paper is twofold: In the first part, we unify and extend the recent developments on honesty theory of perturbed substochastic semigroups (on $L^{1}(\mu )$-spaces or noncommutative $L^{1}$ spaces) to general state spaces; this allows us to capture for instance a honesty theory in preduals of abstract von Neumann algebras or subspaces of duals of abstract $C^{\ast }$-algebras. In the second part of the paper, we provide another honesty theory (a semigroup-perturbation approach) independent of the previous resolvent-perturbation approach and show the equivalence of the two approaches. This second viewpoint on honesty is new even in $L^{1}(\mu )$ spaces. Several fine properties of Dyson-Phillips expansions are given and a classical generation theorem by T. Kato is revisited.

Certain free products of operator spaces

Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO's) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO's. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator spaces.

Asymptotic behavior of Markov semigroups on preduals of von Neumann algebras

We develop a new approach for investigation of asymptotic behavior of Markov semigroup on preduals of von Neumann algebras. With using of our technique we establish several results about mean ergodicity, statistical stability, and constrictiviness of Markov semigroups.

On Fourier algebra homomorphisms

Let G be a locally compact group and let B( G) be the dual space of C ∗(G) , the group C ∗ algebra of G. The Fourier algebra A( G) is the closed ideal of B( G) generated by elements with compact support. The Fourier algebras have a natural operator space structure as preduals of von Neumann algebras. Given a completely bounded algebra homomorphism φ : A(G)→B(H) we show that it can be described, in terms of a piecewise affine map α : Y→G with Y in the coset ring of H, as follows φ(f)= f∘α on Y, 0 off Y when G is discrete and amenable. This extends a similar result by Host. We also show that in the same hypothesis the range of a completely bounded algebra homomorphism φ : A(G)→A(H) is as large as it can possibly be and it is equal to a well determined set. The same description of the range is obtained for bounded algebra homomorphisms, this time when G and H are locally compact groups with G abelian.

Fourier Algebras on Topological Foundation *-Semigroups.

We introduce the notion of the Fourier and Fourier-Stieltjes algebras of a topological *-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras.

Almost Completely Isometric Embeddings between Preduals of von Neumann Algebras

We will prove that if the predual of an injective von Neumann algebra is embedded in the predual of another von Neumann algebra almost completely isometrically, then it is complemented almost completely contractively. This result is the operator space analogue of Dor's theorem.

On the set of finite-dimensional subspaces of preduals of von Neumann algebras

Abstract We prove that the set of all d -dimensional subspaces of preduals of von Neumann algebras is compact in the topology given by the cb-distance.