The Universal Factorization Property for Commutative Subspace Lattices

A a-weakly closed subspace ` of B(' X) is said to have Propett S_ if for any a-weakly closed subspace \5 of a von Neumann algebra -I, {.x E 5 0 A: R-(,(.x) (= ' for all rp E B(0.)) = X ' l, where R , is the right slice map associated with T. It is shown that semidiscrete von Neumann algebras have Property Sa, and various stability properties of the class of a-weakly closed subspaces with Property S0 are established. It is also shown that if (?111, G, a) is a W*-dynamical system such that 'k has Property S0 and C is compact abelian, then all of the spectral subspaces associated with a have Property S0. Some applications of these results to the study of tensor products of spectral subspaces and tensor products of reflexive algebras are given. In particular, it is shown that if L , is a commutative subspace lattice with totally atomic core, and L, is an arbitrary subspace lattice, then alg(i 0 E2) algC1 0 algL?2. Introduction. Slice maps, introduced by Tomiyama, have proved to be a powerful tool in studying tensor products of C*-algebras and von Neumann algebras (see, e.g., [23,24,26-281). In this paper we use slice maps to study tensor products of a-weakly closed subspaces of von Neumann algebras, particularly tensor products of spectral subspaces, and tensor products of reflexive algebras. If ')T and 6. are von Neumann algebras, then for each qp in X*, the predual of .X, there is a unique a-weakly continuous linear map R., from 9DZ 0 9 to 6 such that R,W,(a X b) = (a, q) b (a E 91R, b E DL)l called the right slice map associated with p. Similarly, for each 4 E 9L, there is a left slice map LI, from 9DTh0 6 to 9DT. If 5 C 9Th and Y C % are a-weakly closed subspaces, and 5 8 '5 denotes the a-weak closure of the algebraic tensor product of 5 and C, it is clear that if x E 0 0 then R (x) C 9 for all p E 9<X*. If (x E 5 0 : R ,(x) Efor all qg E ' } = 5 Received by the editors February 19, 1982 and, in revised form, July 20, 1982. 1980 Mathematics Subject Classification. Primary 46L10, 46L55, 46M05, 47D25.

Let C ( H ) C(H) denote the lattice of all (closed) subspaces of a complex, separable Hilbert space H H . Let ( AC) ({\text {AC)}} be the following condition that a subspace lattice F ⊆ C ( H ) \mathcal {F} \subseteq C(H) may or may not satisfy: (AC) \[ F = ϕ ( L ) for some lattice automorphism ϕ of C ( H ) and some commutative subspace lattice L ⊆ C ( H ) . \begin {array}{*{20}{c}} {\mathcal {F} = \phi (\mathcal {L})\;{\text {for}}\;{\text {some}}\;{\text {lattice}}\;{\text {automorphism}}\;\phi \;{\text {of}}\;C(H)} \\ {{\text {and}}\;{\text {some}}\;{\text {commutative}}\;{\text {subspace}}\;{\text {lattice}}\;\mathcal {L} \subseteq C(H).} \\ \end {array} \] Then F \mathcal {F} satisfies ( AC ) ({\text {AC}}) if and only if A ⊆ B \mathcal {A} \subseteq \mathcal {B} for some Boolean algebra subspace lattice B ⊆ C ( H ) \mathcal {B} \subseteq C(H) with the property that, for every K , L ∈ B K,L \in \mathcal {B} , the vector sum K + L K + L is closed. If F \mathcal {F} is finite, then F \mathcal {F} satisfies ( AC ) ({\text {AC}}) if and only if F \mathcal {F} is distributive and K + L K + L is closed for every K , L ∈ F K,L \in \mathcal {F} . In finite dimensions F \mathcal {F} satisfies ( AC ) ({\text {AC}}) if and only if F \mathcal {F} is distributive. Every F \mathcal {F} satisfying ( AC ) ({\text {AC}}) is reflexive. For such F \mathcal {F} , given vectors x , y ∈ H x,y \in H , the solvability of the equation T x = y Tx = y for T ∈ Alg F T \in \operatorname {Alg}\,\mathcal {F} is investigated.

Every infinite tensor product of commutative subspace lattices is unitarily equivalent to a certain lattice of projections on L 2 ( X , ν ) {L^2}(X,\nu ) , where ( X , ν ) (X,\nu ) is an infinite product measure space. This representation reflects the structure of the individual component lattices in that the components of the tensor product correspond to the coordinates of the product space. This result generalizes the similar representation for finite tensor products. It is then used to show that an infinite tensor product of purely atomic commutative subspace lattices must be either purely atomic or noncompact, and in the latter case the algebra of operators under which the lattice is invariant has no compact operators.

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

Compact operators, invariant subspaces, and spectral synthesis

Perturbation theory for groups and lattices

Anoussis and Katsoulis have obtained a criterion for the space $\operatorname {Alg} \mathcal L\cap \mathcal C_p$ to have a closed complement in $\mathcal C_p$, where $\mathcal L$ is a completely distributive commutative subspace lattice. They show that, for a given $\mathcal L$, the set of $p$ for which this complement exists forms an interval whose endpoints are harmonic conjugates. Also, they establish the existence of a lattice $\mathcal L$ for which $\operatorname {Alg} \mathcal L\cap \mathcal C_p$ has no complement for any $p\not =2$. However, they give no specific example. In this note an elementary demonstration of a simple example of this phenomenon is given. From this it follows that for a wide range of lattices $\mathcal L$, $\operatorname {Alg} \mathcal L\cap \mathcal C_p$ fails to have a complement for any $p\not =2$.

Let A be a unital *-algebra. In this paper, under some mild conditions on A, it is shown that a map ? : A ? Ais a nonlinear mixed bi-skew Lie triple derivation if and only if ? is an additive*-derivation. As applications, nonlinear mixed bi-skew Lie triple derivations on prime *-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.

Let [Formula: see text] be a surjective map between some operator algebras such that [Formula: see text] for all [Formula: see text], where [Formula: see text] defined by [Formula: see text] and [Formula: see text] is Jordan product, i.e. [Formula: see text]. In this paper, we determine the concrete form of map [Formula: see text] on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if [Formula: see text] is a factor von Neumann algebra that satisfies [Formula: see text] for all [Formula: see text] and idempotents [Formula: see text] then there exists nonzero scalar [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]

Nonlinear maps preserving Lie products on factor von Neumann algebras

Let [Formula: see text] be a factor von Neumann algebra with dim[Formula: see text]. In this paper, we prove that a map [Formula: see text] satisfies [Formula: see text] for all [Formula: see text] if and only if there exists a real number [Formula: see text] such that [Formula: see text], where [Formula: see text] is an additive ∗-derivation and [Formula: see text].

Given a T0 paratopological group G and a class C of continuous homomorphisms of paratopological groups, we define the C-semicompletionC[G) and C-completionC[G] of the group G that contain G as a dense subgroup, satisfy the T0-separation axiom and have certain universality properties. For special classes C, we present some necessary and sufficient conditions on G in order that the (semi)completions C[G) and C[G] be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group G whose C-semicompletion C[G) fails to be a T1-space, where C is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group G contains an ω-bounded sequentially compact subgroup H such that H is a topological group but its closure in G fails to be a subgroup.

When formulating universal properties for objects in a dagger category, one usually expects a universal property to characterize the universal object up to unique unitary isomorphism. We observe that this is automatically the case in the important special case of C$^*$-categories, provided that one uses enrichment in Banach spaces. We then formulate such a universal property for infinite direct sums in C$^*$-categories, and prove the equivalence with the existing definition due to Ghez, Lima and Roberts in the case of W$^*$-categories. These infinite direct sums specialize to the usual ones in the category of Hilbert spaces, and more generally in any W$^*$-category of normal representations of a W$^*$-algebra. Finding a universal property for the more general case of direct integrals remains an open problem.

We establish some properties of quantum quasi-shuffle algebras. They include the necessary and sufficient condition for the construction of the quantum quasi-shuffle product, the universal property, and the commutativity condition. As an application, we use the quantum quasi-shuffle product to construct a linear basis of $T(V)$, for a special kind of Yang-Baxter algebras $(V,m,\sigma)$.

In asymmetrically warped spacetimes different warp factors are assigned to\nspace and to time. We discuss causality properties of these warped brane\nuniverses and argue that scenarios with two extra dimensions may allow for\ntimelike curves which can be closed via paths in the extra-dimensional bulk. In\nparticular, necessary and sufficient conditions on the metric for the existence\nof closed timelike curves are presented. We find a six-dimensional warped\nmetric which satisfies the CTC conditions, and where the null, weak and\ndominant energy conditions are satisfied on the brane (although only the former\nremains satisfied in the bulk). Such scenarios are interesting, since they open\nthe possibility of experimentally testing the chronology protection conjecture\nby manipulating on our brane initial conditions of gravitons or hypothetical\ngauge-singlet fermions (sterile neutrinos) which then propagate in the extra\ndimensions.\n