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1.
We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities
are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe
the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands
in this algebra. We prove that if a function from ℝ+ into ℝ+ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on
the set of positive self-adjoint operators affiliated with this algebra. 相似文献
2.
3.
A. I. Shtern 《Functional Analysis and Its Applications》2003,37(2):157-159
Proofs of two assertions are sketched. 1) If the Banach space of a von Neumann algebra A is the third dual of some Banach space, then the space A is isometrically isomorphic to the second dual of some von Neumann algebra A and the von Neumann algebra A is uniquely determined by its enveloping von Neumann algebra (up to von Neumann algebra isomorphism) and is the unique second predual of A (up to isometric isomorphism of Banach spaces). 2) An infinite-dimensional von Neumann algebra cannot have preduals of all orders. 相似文献
4.
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 selfadjoint part of the predual is 1-Plichko as well. 相似文献
5.
A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a
Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra
and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter
family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection
lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra
of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed
elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements
to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice,
and give several sufficient conditions for modularity of the projection lattice. 相似文献
6.
We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on ? are isomorphic even when the Continuum Hypothesis fails. 相似文献
7.
We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi‐splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi‐splitting subspaces are non‐equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi‐splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side. 相似文献
8.
The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for L2(Rd) if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for II1 factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when G is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well-known problem about the classification of free group von Neumann algebras. 相似文献
9.
We prove that a C
*-algebra A or a predual N
* of a von Neumann algebra N has the Daugavet property if and only if A (or N) is non-atomic. We also prove a similar (although somewhat weaker) result for non-commutative L
p-spaces corresponding to non-atomic von Neumann algebras. 相似文献
10.
Yasuhide Miura 《Proceedings of the American Mathematical Society》1996,124(8):2475-2478
The purpose of this paper is to prove that a completely positive projection on a Hilbert space associated with a standard form of a von Neumann algebra induces the existence of a conditional expectation of the von Neumann algebra with respect to a normal state, and we consider the application to a standard form of an injective von Neumann algebra.
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12.
Maria Joiţa 《Rendiconti del Circolo Matematico di Palermo》2002,51(1):83-94
In this paper, we will prove some properties of locally von Neumann algebras. In particular, we will show that every locally
von Neumann algebra is the dual of a certain locally convex space and also, we will show the existence of a polar decomposition
for every element in a locally von Neumann algebra. 相似文献
13.
Order - We define and study an alternative partial order, called the spectral order, on a synaptic algebra—a generalization of the self-adjoint part of a von Neumann algebra. We prove that if... 相似文献
14.
We study the relationship between (o)-convergence and almost-everywhere convergence in the Hermite part of the ring of unbounded measurable operators associated with a finite von Neumann algebra. In particular, we prove a theorem according to which (o)-convergence and almost-everywhere convergence are equivalent if and only if the von Neumann algebra is of the type I. 相似文献
15.
令H是有限维Hopf代数,A是左H-模代数。本文证明了A是Gorenstein代数的充分必要条件。A^H也是Gorenstein代数的条件。它是Enochs EE,GarciaJJ和del RioA关于群作用相应的理论的推广,同时给出A/A^H是Frobenius扩张的条件。 相似文献
16.
O. E. Hentosh 《Ukrainian Mathematical Journal》2009,61(7):1075-1092
A compatibly bi-Hamiltonian Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems is obtained by using a
relation for the Casimir functionals of the central extension of a Lie algebra of superconformal even vector fields of two
anticommuting variables. Its matrix Lax representation is determined by using the property of the gradient of the supertrace
of the monodromy supermatrix for the corresponding matrix spectral problem. For a supersymmetric Laberge–Mathieu hierarchy,
we develop a method for reduction to a nonlocal finite-dimensional invariant subspace of the Neumann type. We prove the existence
of a canonical even supersymplectic structure on this subspace and the Lax–Liouville integrability of the reduced commuting
vector fields generated by the hierarchy. 相似文献
17.
令M_1为一个有限的von Neumann代数,τ_1为其上的一个忠实正规迹态.我们将证明,如果M_1中存在一列两两正交的酉元列{u_k:k∈N},则对任意具有忠实正规迹态τ_2的有限von Neumann代数M_2(≠C),迹自由积(M_1,τ_1)*(M_2,τ_2)是Ⅱ_1型因子.作为推论可以得出,如果M_1有一个von Neumann子代数N不包含最小投影,则对任意具有忠实迹态τ_2的有限von Neumann代数M_2(≠C),迹自由积(M_1,τ_1)*(M_2,τ_2)是Ⅱ_1型因子. 相似文献
18.
M. M. Ibragimov K. K. Kudaibergenov Zh. Kh. Seipullaev 《Russian Mathematics (Iz VUZ)》2018,62(5):27-33
We prove that predual of real part of von Neumann algebra is strongly facially symmetric space if and only if is it a direct sum of Abelian algebra and algebra of I2 type. At that, neutral strongly facially symmetric space is predual to Abelian algebra, only. 相似文献
19.
Maryam H.A. Al-Rashed Bogus?aw Zegarliński 《Linear algebra and its applications》2011,435(12):2999-3013
We introduce and study the noncommutative Orlicz spaces associated to a normal faithful state on a semifinite von Neumann algebra. We also prove some convergence theorems for the (unbounded) trace measurable operators. 相似文献
20.
Jaeseong Heo 《Journal of Mathematical Analysis and Applications》2010,365(1):308-314
In this paper we study the transitive algebra question by considering the invariant subspace problem relative to von Neumann algebras. We prove that the algebra (not necessarily ∗) generated by a pair of sums of two unitary generators of L(F∞) and its commutant is strong-operator dense in B(H). The relations between the transitive algebra question and the invariant subspace problem relative to some von Neumann algebras are discussed. 相似文献