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This paper deals with properties of the ultraproducts for various structures. We introduce and study the concept of the ergodic action of a group with respect to a normal state on an abelian von Neumann algebra. In particular, we provide an example showing that the ultraproduct of ergodic states, generally speaking, is not ergodic. The ultraproduct of the Radon measures on a compact convex subset of a locally convex space is also investigated in the paper. As is well-known, the study of the extreme points in the state set for a $$C^{*}-$$ algebra is a very interesting problem in itself. Considering the ultraproducts of $$C^{*}$$ -algebras and the states on these algebras, we get quite nontrivial results.

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio&rsquo, s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.

Let $$\mathcal {A}$$ and $$\mathcal {B}$$ be two von Neumann algebras with no central abelian projections. In this paper, it is proved that if a not necessarily linear bijective map $$\Phi :\mathcal {A}\rightarrow \mathcal {B}$$ satisfies $$\Phi (A^{*}B+B^{*}A)=\Phi (A)^{*}\Phi (B)+\Phi (B)^{*}\Phi (A)$$ for all $$A, B\in \mathcal {A}$$ , then $$\Phi $$ is a sum of a linear $$*$$ -isomorphism and a conjugate linear $$*$$ -isomorphism.

We prove an Ambrosetti-Prodi type result for a Neumann problem associated to the equation \begin{document}$u''+f(x, u(x))=μ$\end{document} when the nonlinearity has the following form: \begin{document}$f(x, u):=a(x)g(u)-p(x)$\end{document} . The assumptions considered generalize the classical one, \begin{document}$f(x, u)\to+∞$\end{document} as \begin{document}$|u|\to+∞$\end{document} , without requiring any uniformity condition in \begin{document}$x$\end{document} . The multiplicity result which characterizes these kind of problems will be proved by means of the shooting method.

In this note, we prove that many von Neumann algebras can be generated by two unitary operators u and v with u 2 = v 3 = 1 . As applications, three important classes of II1 factors possessing this property are those with Property Γ, or with a Cartan masa, or non-prime ones.

Assume that T is a linear relation acting between two real or complex Hilbert spaces and , respectively. T is shown to be closed if and only if the linear relations and are both selfadjoint linear relations on the corresponding Hilbert spaces and , respectively.

We prove that every weak-2-local derivation on a finite von Neumann algebra is a derivation. We also show that every weak-2-local derivation on a simple unital C-algebra with a separating family of...

We investigate boundary representations in the context where Hilbert spaces are replaced by $$\hbox {C}^{*}$$ -modules over abelian von Neumann algebras and apply this to study $$\hbox {C}^{*}$$ -extreme points. We present an (unexpected) example of a weak* compact $$\mathcal {B}$$ -convex subset of $${\mathbb {B}}(\mathcal {H})$$ without $$\mathcal {B}$$ -extreme points, where $$\mathcal {B}$$ is an abelian von Neumann algebra on a Hilbert space $$\mathcal {H}$$ . On the other hand, if $$\mathcal {A}$$ is a von Neumann algebra with a separable predual and whose finite part is injective, we show that each weak* compact $$\mathcal {A}$$ -convex subset of $$\ell ^{\infty }(\mathcal {A})$$ is generated by its $$\mathcal {A}$$ -extreme points.

Let A be a factor von Neumann algebra acting on a complex Hilbert space H with dim ⁡ ( A ) ≥ 2 . We show that each local Lie derivation from A into itself is a Lie derivation.

The main purpose of this paper is to prove that every generalized derivation on a commutative von Neumann algebra is identically zero.

The harmonic Neumann function is constructed for a class of hyperbolic strips inside the unit disc via the parqueting-reflection principle. As it turns out this Neumann function is related to the respective Green function, see [the second author, Green function for a hyperbolic strip and a class of related plane domains, Appl. Anal. 93(2014), 2370–2385], in the same way as for the case of e.g. the unit disc or half planes, etc. On this basis the Neumann problem for the Poisson equation is solved in an explicit way. Such explicit solutions serve for applications as in engineering or mathematical physics.

In this paper, we give an answer to a conjecture due to Muscalu. We also prove a non-commutative analogue of Cwikel’s theorem.

This article investigates a bijective map $\Phi$ between two von Neumann algebras, one of which has no central abelian projections, satisfying $\Phi([[A,B]_{*},C]_{*})=[[\Phi(A),\Phi(B)]_{*},\Phi(C)]_{*}$ for all $A,B,C$ in the domain, where $[A,B]_{*}=AB-BA^{\ast}$ is the skew Lie product of $A$ and $B$ . We show that the map $\Phi(I)\Phi$ is a sum of a linear $*$ -isomorphism and a conjugate linear $*$ -isomorphism, where $\Phi(I)$ is a self-adjoint central element in the range with $\Phi(I)^{2}=I$ .

A state-preserving automorphism of a von Neumann algebra induces a canonical unitary operator on the GNS Hilbert space of the state which fixes the vacuum. This unitary commutes with both the modular operator of the state and its modular conjugation. We prove an extension of this result for state-preserving unital completely positive maps.

Let M be a von Neumann algebra equipped with a normal finite faithful normalized trace τ, and let A be a tracial subalgebra of M. Let E be a symmetric quasi-Banach space on [0, 1]. We show that A has an LE(M)-factorization if and only if A is a subdiagonal algebra.

We show the following function on m-tuples of positive operators in finite von Neumann algebra M : Φ p ( A 1 , A 2 , … , A m ) = ‖ ( ∑ k = 1 m A k p ) 1 p ‖ L E ( M ) is jointly convex for 1 ≤ p ≤ 2 and for any symmetric norm ‖ . ‖ L E ( M ) associated with M .

A new characterization of commutative von Neumann regular rings is given in terms of linear control systems having, locally, a Brunovsky Canonical Form. The problem of enumerating reachable systems over certain von Neumann regular ring is solved extending classical results over fields.

A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of distinguished gradient flow lines of the eigenfunction - those which are connected to saddle points. These give rise to Neumann domains. We establish complementary definitions for Neumann domains and Neumann lines and use basic Morse homology to prove their fundamental topological properties. We study the eigenfunction restrictions to these domains. Their zero set, critical points and spectral properties allow to discuss some aspects of counting the number of Neumann domains and estimating their geometry.

This work contains some structural results for von Neumann algebras arising from measure preserving actions by direct products of groups on probability spaces. The technology and the methods we use are a continuation of those used by Chifan and Sinclair in [Ionut Chifan and Thomas Sinclair. On the structural theory of II1 factors of negatively curved groups, ArXiv e-prints, March 2011]. By employing these methods, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. We show for instance that every II1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid [Narutaka Ozawa. Solid von Neumann algebras, Acta Math., 192 (2004), 111–117]. We also obtain a product version of this result: any maximal abelian *-subalgebra of any II1 factor associated with a finite direct product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with Ioana’s cocycle superrigidity theorem [Adrian Ioana. Cocycle superrigidity for profinite actions of property (T) groups, Duke Math. J., to appear ], we prove that compact actions by finite products of lattices in Sp(n, 1), n ≥ 2, are virtually W ∗-superrigid. The results presented here are joint work with Ionut Chifan and Thomas Sinclair. They constitute the substance of an article which has already been submitted for publication [Ionut Chifan, Thomas Sinclair, and Bogdan Udrea. On the structural theory of II1 factors of negatively curved groups, ii. actions by product groups, ArXiv e-prints, August 2011].

In this paper, we prove that every \(*\)-Lie derivable mapping on a von Neumann algebra with no central abelian projections can be expressed as the sum of an additive \(*\)-derivation and a mapping with image in the center vanishing at commutators.

Let and be two factor von Neumann algebras with dimensions greater than 1. In this paper, we consider a map from onto preserving the -Lie product, that is, for all . It is proved that a bijective map from onto preserves the -Lie product, if and only if is a linear -isomorphism or a conjugate linear -isomorphism. In particular, if the factor von Neumann algebras and are and , then is a unitary isomorphism or a conjugate unitary isomorphism.

The Hecke algebra $\mathbb{C}_q[W]$ of a Coxter group $W$, associated to parameter $q$, can be completed to a von Neumann algebra $\mathcal{N}_q(W)$. We study such algebras in case where $W$ is right-angled. We determine the range of $q$ for which $\mathcal{N}_q(W)$ is a factor, i.e. has trivial center. Moreover, in case of nontrivial center, we prove a result allowing to decompose $\mathcal{N}_q(W)$ into a finite direct sum of factors., Comment: 15 pages, 5 figures

Let be a factor von Neumann algebra. Suppose that is a nonlinear skew Jordan derivable map. Then, is an additive -derivation. In particular, if the von Neumann algebra is infinite type I factors, a concrete characterization of is given.

In this paper, we discuss the dual of a von Neumann–Schatten p-frames in separable Banach spaces and obtain some of their characterizations. Moreover, we present a classical perturbation result to von Neumann–Schatten p-frames.

Let be a semi-finite von Neumann algebra factor with dimension greater than 4. We prove that every 2-local Lie derivation of can be written in the form for all , where T is an element in , and is a homogeneous map satisfying for every and every commutator .

Let (M,τ) be a noncommutative probability space, (Mn)n≥1 a sequence of von Neumann subalgebras of M and N a von Neumann subalgebra of M. We introduce the notions of μ-approach and orthogonal approach for (Mn)n≥1 and prove that ɛ(x|Mn)→Lpɛ(x|N) for any x∈Lp(M)(1≤p

We prove that every derivation acting on a von Neumann algebra M with values in a quasi-normed bimodule of locally measurable operators affiliated with M is necessarily inner.

Let be a factor von Neumann algebra and be the *-Jordan derivation on , that is, for every , where , then is additive *-derivation.

We obtain new necessary and sufficient commutation conditions for nonnegative operators and projections in terms of operator inequalities. It is shown that in the general case in this inequalities the projections cannot be replaced by arbitrary nonnegative operators with preservation of operators commutativity. We also present new necessary and sufficient commutation conditions for projections in terms of operator inequalities. These inequalities are applied for trace characterization on von Neumann algebras in the class of all positive normal functionals. We also consider the following problems: I. Characterization of traces among arbitrary weights on von Neumann algebras. II. Characterization of tracial functionals among all positive linear functionals on C∗-algebras. III. Characterization of commutativity for C∗-algebras.

Given a system ( A , B ) over a commutative von Neumann regular ring R, it is proved that there exist a matrix F and a vector u such that the single-input system ( A + B F , B u ) and the original system ( A , B ) have the same module of reachable states and the same set of polynomials assignable by state feedback. Moreover, there is a bijection between reachable states and assignable polynomials, in the form of a certain isomorphism of R-modules, and the existence of this isomorphism for all systems actually characterizes von Neumann regular rings. Finally, the set of assignable polynomials to a single-input system is completely described for arbitrary commutative rings, which in the case of von Neumann regular rings completes the study of assignable polynomials to multi-input systems.

We study measures on orthoideals in the set of projections in von Neumann algebra, which appear within a framework of L1-spaces associated with positive operators in the algebra. We show that there exists a positive measure on orthoideal, which cannot be extended to a normal weight on the algebra, but can be represented as a difference of two positive measures extendable to normal weights.

Let L be a double triangle lattice of projections in a finite von Neumann algebra acting on a separable and complex Hilbert space K . We show that every derivation from the reflexive algebra determined by L into B ( K ) is quasi-spatial and automatically continuous. We also obtain that every local derivation on the reflexive algebra is a derivation.

We discuss the fact that there is a crucial contradiction within Von Neumann’s theory. We derive a proposition concerning a quantum expected value under an assumption of the existence of the orientation of reference frames in N spin-1/2 systems (1 ≤ N < +∞). This assumption intuitively depictures our physical world. However, the quantum predictions within the formalism of Von Neumann’s projective measurement violate the proposition with a magnitude that grows exponentially with the number of particles. We have to give up either the existence of the directions or the formalism of Von Neumann’s projective measurement. Therefore, Von Neumann’s theory cannot depicture our physical world with a violation factor that grows exponentially with the number of particles. The theoretical formalism of the implementation of the Deutsch-Jozsa algorithm relies on Von Neumann’s theory. We investigate whether Von Neumann’s theory meets the Deutsch-Jozsa algorithm. We discuss the fact that the crucial contradiction makes the quantum-theoretical formulation of Deutsch-Jozsa algorithm questionable. Further, we discuss the fact that projective measurement theory does not meet an easy detector model for a single Pauli observable. Especially, we systematically describe our assertion based on more mathematical analysis using raw data. We propose a solution of the problem. Our solution is equivalent to changing Planck’s constant () to a new constant ( ) 2  . It may be said that a new type of the quantum theory early approaches Newton’s theory in the macroscopic scale than the old quantum theory does. We discuss how our solution is used in an implementation of Deutsch’s algorithm.

Let τ be a tracial normal state on a von Neumann algebra, L 1 ( τ ) be the space of integrable self-adjoint operators, and S be the space of self-adjoint measurable operators. We prove that every positive linear operator from an ordered Banach space to S can be factorized through L 1 ( τ ) .

Let $\mathscr{R}$ be a finite von Neumann algebra with a faithful tracial state $\tau $ and let $\Delta$ denote the associated Fuglede-Kadison determinant. In this paper, we characterize all unital bijective maps $\phi$ on the set of invertible positive elements in $\mathscr{R}$ which satisfy $$\Delta(\phi(A)+\phi(B)) = \Delta(A+B).$$ We show that any such map originates from a $\tau$-preserving Jordan $*$-automorphism of $\mathscr{R}$ (either $*$-automorphism or $*$-anti-automorphism in the more restrictive case of finite factors). In establishing the aforementioned result, we make crucial use of the solutions to the equation $\Delta(A + B) = \Delta(A) + \Delta(B)$ in the set of invertible positive operators in $\mathscr{R}$. To this end, we give a new proof of the inequality $$\Delta(A+B) \ge \Delta(A) + \Delta(B),$$ using a generalized version of the Hadamard determinant inequality and conclude that equality holds for invertible $B$ if and only if $A$ is a nonnegative scalar multiple of $B$., Comment: 11 pages

Let $\mathcal {A}$ be a factor von Neumann algebra, with unit $ I $, which contains a nontrivial projection $ P_{1} $, and let $\psi :\mathcal {A} \rightarrow \mathcal {A}$ be a surjective map that satisfies one of the two conditions: $\psi (A)\psi (P) + \lambda \psi (P)\psi (A) = AP + \lambda PA$ and $\psi (A)\psi (P) + \lambda \psi (P)\psi (A)^{\ast } = AP + \lambda PA^{\ast }$ for all $A \in \mathcal {A}$ and $P \in \lbrace P_{1}, I - P_{1}\rbrace $ and $\lambda \in \lbrace -1, 1\rbrace $. Then, we determine the concrete form of $\psi $.

Let F′ be the commutator subgroup of F and let Γ0 be the cyclic group generated by the first generator of F. We continue the study of the central sequences of the factor L(F′), and we prove that the abelian von Neumann algebra L(Γ0) is a strongly singular MASA in L(F). We also prove that the natural action of F on [0, 1] is ergodic and that its ratio set is {0} ∪ {2k; k ∞ Z}.

A distance between von Neumann algebras is introduced, depending on a further norm inducing the $w^*$-topology on bounded sets. Such notion is related both with the Gromov-Hausdorff distance for quantum metric spaces of Rieffel and with the Effros-Marechal topology on the von Neumann algebras acting on a Hilbert space. This construction is tested on the local algebras of free quantum fields endowed with norms related with the Buchholz-Wichmann nuclearity condition, showing the continuity of such algebras w.r.t. the mass parameter., Comment: 22 pages, no figures. To apper in the Journal of Functional Analysis