Order type relations on the set of tripotents in a JB∗-triple

On products of symmetries in von Neumann algebras

It was proved by Fillmore that a unitary of a properly infinite von Neumann algebra A can be expressed as a product of at most four symmetries. In this paper we introduce an axiom (ENCP) for Baer $^ \ast$-rings and prove that Fillmore’s result is true if A is a properly infinite Baer $^ \ast$-ring satisfying (ENCP) and $LP \sim RP$. This also affirmatively answers the open problem on $A{W^ \ast }$-algebras posed by Berberian.

It was proved by Fillmore that a unitary of a properly infinite von Neumann algebra A can be expressed as a product of at most four symmetries. In this paper we introduce an axiom (ENCP) for Baer *-rings and prove that Fillmore's result is true if A is a properly infinite Baer *-ring satisfying (ENCP) and LP RP. This also affirmatively answers the open problem on AW*algebras posed by Berberian.

If a semicircular element and the diagonal subalgebra of a matrix algebra are free in a finite von Neumann algebra (with respect to a normal trace), then, up to the conjugation by a diagonal unitary element, all entries of the semicircular element are uniquely determined in the sense of (joint) distribution. Suppose a selfadjoint element is free with the diagonal subalgebra. Then, in the matrix decomposition of the selfadjoint element, any two entries cannot be free with each other unless the selfadjoint element is semicircular. We also define a “matricial distance” between two elements and show that such distance for two free semicircular elements in a finite von Neumann algebra is nonzero and independent of the properties of the von Neumann algebra.

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group G/ζ(G) of a group G is finite, then its derived subgroup [G,G] is also finite. This theorem was proved by B. Neumann in 1951. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, n-groups, associative algebras, Lie algebras, Lie n-algebras. In 2016, L.A. Kurdachenko, J. Otal and O.O. Pypka proved an analogue of Schur theorem for Leibniz algebras: if central factor-algebra L/ζ(L) of Leibniz algebra L has finite dimension, then its derived ideal [L,L] is also finite-dimensional. Moreover, they also proved a slightly modified analogue of Schur theorem: if the codimensions of the left ζ^l (L) and right ζ^r (L) centers of Leibniz algebra L are finite, then its derived ideal [L,L] is also finite-dimensional. One of the generalizations of Leibniz algebras is the so-called Leibniz n-algebras. It is worth noting that Leibniz n-algebra theory is currently much less developed than Leibniz algebra theory. One of the directions of development of the general theory of Leibniz n-algebras is the search for analogies with other types of algebras. Therefore, the question of proving analogs of the above results for this type of algebras naturally arises. In this article, we prove the analogues of the two mentioned theorems for Leibniz n-algebras for the case n=3. The obtained results indicate the prospects of further research in this direction.

This chapter serves as an overview of some of the basic building blocks for quantum probability, quantum Markov semigroups/processes, and their large time asymptotic behavior that are to follow. We start out with a brief review of complex Hilbert spaces and their topological dual spaces together with the concepts of weak and strong convergence. The concepts of linear operators on complex Hilbert spaces are introduced. Special classes of bounded linear operators including self-adjoint, Hilbert-Schmidt, trace-class, compact and projection operators, operator-valued spectral measures, and the celebrated spectral representation theorem due originally to von Neumann (see von Neumann [vNeu55]) are discussed. We also define various concepts of operator topologies, such as norm-topology, strong topology, weak topology, σ -strong topology, σ -weak topology, and weak*-topology, on the space of bounded linear operators. Equivalence of some of these topologies under appropriate conditions are illustrated. This chapter also introduces the 2 major types of algebras, namely, the C* -algebra and von Neumann algebra of operators on a complex Hilbert space. These 2 different types of algebras are all to be denoted by A . However, the results will be stated with specification to which of the algebras is under consideration. Unless otherwise stated, it is assumed throughout the book that all algebras are unital; i.e., they contain the identity operator. These algebras, especially the von Neumann algebra, are important tools for describing quantum probability spaces and quantum systems. Many of the results presented in this chapter are stated in terms of C* -algebras in general without specifications to the von Neumann algebra. One of the important topology on A that plays an important role in studying quantum Markov semi-groups or quantum Markov processes is the so called σ -weak continuity. Finally, we define representation of a C* -algebra and prepare the background material for describing Gelfand-Naimark-Segal construction for a representation of C* -algebra, which is described in detail.

Let ( x i ) (x_{i}) be a finite collection of commuting self-adjoint elements of a von Neumann algebra M \mathcal {M} . Then within the (abelian) C*-algebra they generate, these elements have a least upper bound x x . We show that within M \mathcal {M} , x x is a minimal upper bound in the sense that if y y is any self-adjoint element such that x i ≤ y ≤ x x_{i} \leq y \leq x for all i i , then y = x y = x . The corresponding assertion for infinite collections ( x i ) (x_{i}) is shown to be false in general, although it does hold in any finite von Neumann algebra. We use this sort of result to show that if N ⊂ M \mathcal {N} \subset \mathcal {M} are von Neumann algebras, Φ : M → N \Phi : \mathcal {M} \to \mathcal {N} is a faithful conditional expectation, and x ∈ M x \in \mathcal {M} is positive, then Φ ( x n ) 1 / n \Phi (x^{n})^{1/n} converges in the strong operator topology to the “spectral order majorant” of x x in N \mathcal {N} .

We consider maximization of the relative entropy (with respect to a fixed normal state) in a von Neumann algebra among the states having fixed expectation for finitely many self-adjoint elements.

We analyze a certain class of von Neumann algebras generated by selfadjoint elements \(\), for \(\) satisfying the general commutation relations: $$$$ Such algebras can be continuously embedded into some closure of the set of finite linear combinations of vectors \(\), where \(\) is an orthonormal basis of a Hilbert space \(\). The operator which represents the vector \(\) is denoted by \(\) and called the “Wick product” of the operators \(\). We describe explicitly the form of this product. Also, we estimate the operator norm of \(\) for \(\). Finally we apply these two results and prove that under the assumption \(\) all the von Neumann algebras considered are II 1 factors.

The theories of weights, traces and states are often referred as non commutative integration. If the von Neumann algebra in question is abelian, then our theory is precisely the theory of measures and integration. In fact, the weight value of a self-adjoint element is given precisely by the integration of the corresponding function on the spectrum relative to the measure corresponding to the weight. As there are many non-commuting self-adjoint elements in the algebra, we have to consider various spectral measures even if we fix one weight and we can not represent non-commuting self-adjoint elements as functions on the same space. The striking difference between the commutative case and the non-commutative case is the appearance of one parameter automorphism group which is determined by the weight. Namely, weights and/or states determine the dynamics of the system which does not have the commutative counter part. We have explored the relationship between weights and the modular automorphism groups so far. We now further investigate how the dynamics, i.e. the modular automorphism groups, of the algebra relate the different spaces associated with the algebras. First, we study the underlying Hilbert space of the algebra and find the intrinsic pointed convex cone there, which is called the natural cone, in the first section. The theory developed there allows us to view the standard Hilbert space as the square root of the predual of the algebra as well as to represent the automorphism group Aut(M), of a von Neumann algebra M, as the group of unitaries which leaves the natural cone globally invariant.

Let A and B be two von Neumann algebras. For A,B ? A, define by [A,B]* = AB-BA* and A ? B = AB + BA* the new products of A and B. Suppose that a bijective map ? : A ? B satisfies ?([A ? B,C]*) = [?(A)? ?(B),?(C)]* for all A,B,C ? A. In this paper, it is proved that if A and B be two von Neumann algebras with no central abelian projections, then the map ?(I)? is a sum of a linear *-isomorphism and a conjugate linear +-isomorphism, where ?(I) is a self-adjoint central element in B with ?(I)2 = I. If A and B are two factor von Neumann algebras, then ? is a linear *-isomorphism, or a conjugate linear *-isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.

Using approximations, we give several characterizations of separability of bimodules. We also discuss how separability properties can be used to transfer some representation theoretic properties from one ring to another: contravariant finiteness of the subcategory of (finitely generated) left modules with finite projective dimension, finitistic dimension, finite representation type, Auslander algebra, tame or wild representation type.

Understanding, discovering, and proving useful properties of sophisticated data structures are central problems in program verification. A particularly challenging exercise for shape analyses involves reasoning about sophisticated shape transformers that preserve the shape of a data structure (e.g., the data structure skeleton is always maintained as a balanced tree) or the relationship among values contained therein (e.g., the in-order relation of the elements of a tree or the parent-child relation of the elements of a heap) across program transformations. In this talk, we consider the specification and verification of such transformers for ML programs. The structural properties preserved by transformers can often be naturally expressed as inductively-defined relations over the recursive structure evident in the definitions of the datatypes they manipulate. By carefully augmenting a refinement type system with support for reasoning about structural relations over algebraic datatypes, we realize an expressive yet decidable specification language, capable of capturing useful structural invariants, which can nonetheless be automatically verified using off-the-shelf type checkers and theorem provers. Notably, our technique generalizes to definitions of parametric relations for polymorphic data types which, in turn, lead to highly composable specifications over higher-order polymorphic shape transformers.

We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra some standard results concerning the spectral properties exhibited by temperature states of disordered quantum spin systems. We investigate the Arveson spectrum, known to physicists as the set of the Bohr frequencies. We also establish its connection with the Connes and Borchers spectra, and with the associated invariants for such W ∗-dynamical systems which determine the type of von Neumann algebras generated by a temperature state. We prove that all such spectra are independent of the disorder. Such results cover infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice ℤ d , including models exhibiting some standard spin-glass-like behavior. As a natural application, we show that a temperature state can generate only a type $\mathop {\rm {III}}$ von Neumann algebra (with the type $\mathop {\rm {III_{0}}}$ component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type $\mathop {\rm {III_{\lambda }}}$ for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.

A class of von Neumann algebras associated with the normal representation (i.e. the Fock representation) of the canonical commutation relations has been studied in an earlier paper. > The von Neumann algebra in question is always the tensor product of an abelian algebra and a factor of the infinite type. A necessary and sufficient condition for the factor to be of type Icc has been obtained in reference 1). In the present paper it will be shown that the factor in question is of type IIIcc unless it is of type t'. Recently there have been some interests in the type of von Neumann algebras of local observables in quantum field theory. J,SJ,), ) The von Neumann algebras of 'local observables for a free scalar field) as well as those for a generalized free field belong to the class of von Neumann algebras considered in reference 1) and our result implies that they are either type III' or t:o. In particular, for those cases where type Icc have been excluded,J,sl, l the algebra must be of type IIICXJ . Our result also implies that the algebra of all creation and annihilation operators in quantum theory of an infinite free Bose gasl is a factor of type IIIcc when no condensation is present.