The slice map problem for $\sigma $-weakly closed subspaces of von Neumann algebras

We present a necessary and sufficient condition for a commutative subspace lattice (CSL) to have the property that every sublattice has the universal factorization property (UFP). Then we derive a sufficient condition for a CSL to have the UFP which applies to all known examples of CSL’s with the UFP. (The preceding results are, in fact, obtained for CSL- subalgebras of factor von Neumann algebras.) Finally, we give an example of a CSL which has the UFP, even though it is not a direct sum of a complemented lattice and a CSL with the property that every linearly ordered sublattice is countable.

Let H_{\mathfrak c} be the incomplete infinite tensor product of Hilbert spaces H_⍳ containing a product vector \otimes x_⍳ , where \mathfrak c denotes the equivalence class of the \mathfrak C_0 -sequence \{x_⍳\} . Let E_{\mathfrak c} be the projection on H_{\mathfrak c} in the complete infinite tensor product H of H_⍳ . Let \mathfrak R be the von Neumann algebra on H generated by von Neumann algebra \mathfrak R_⍳ on H_⍳ and E(\mathfrak c) be the central support of E_{\mathfrak c} in \mathfrak R' . Two \mathfrak C_0 -sequences \{x_⍳\} and \{y_⍳\} , and their equivalence classes \mathfrak c and \mathfrak c' , are defined to be p -equivalent if there exist partial isometries p_⍳\in \mathfrak R'_⍳ such that \{ x_⍳\} and \{p_⍳y_⍳\} are equivalent and p^*_⍳p_⍳y_⍳= y_⍳ . They are defined to be _ u -equivalent if p_⍳ can be chosen unitary. We prove that E(\mathfrak c) is the sum of E_{\mathfrak c'} with \mathfrak c' , p -equivalent to \mathfrak c . If the index set is countable, p -equivalence and u -equivalence coincide.

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.

Operator algebras come in many flavours. For the purpose of this article, however, the term is only used for one of two kinds of self-adjoint algebras of operators on Hilbert space, viz., C*- algebras (which are norm-closed) or von Neumann algebras (which are closed in the topology of pointwise strong convergence or equivalently, in the weak-* topology it inherits as a result of being a Banach dual space). To be fair, there are a number of people in India (eg., Gadadhar Misra, Tirthankar Bhattacharyya, Jaydeb Sarkar, Santanu Dey, etc.,) who work on non-selfadjoint algebras, mostly from the point of view of connections with complex function theory; but in the interest of restricting the size of this paper, I confine myself here to selfadjoint algebras. I apologise for ways in which my own personal taste and limitations colour this depiction of operator algebras. Another instance of this arbitrary personal taste is a decision to concentrate on the work of younger people. Thus, the work of the more senior people who have worked in operator algebras is only seen via their collaborations with younger people: e.g., KRP via Srinivasan and Rajarama Bhat, Kalyan Sinha via Debashish, Partha, Arup, Raja, etc,. and me via Vijay, Srinivasan and Panchugopal. Not long ago, interest in operator algebras in India was restricted to the three centres of the Indian Statistical Institute. Now, I am happy to note that it has spread to IMSc, some IITs, IISERs, NISER, JNU,.... My role in this article has been merely that of compiling inputs from many active Indian operator algebraists that came to my mind. I wrote soliciting a response from a certain number of them, then put together the responses received. (I apologise to those people who were omitted in this process). My colleague, Partha, with the help of his collaborator Arup, agreed to take care of the C*-related inputs, while I take care of the von Neumann-related ones with the help of my collaborator Vijay. What follows are some areas of ongoing research done in von Neumann algebras in India and some names of people doing such work: (a) subfactors and planar algebras, (Vijay Kodiyalam of IMSc, Chennai); (b) quantum dynamical systems and complete positivity (Rajarama Bhat of ISI, Bengaluru); (c) E0 semigroups (R. Srinivasan of CMI, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar), and (d) Masas in II1 von Neumann algebras and free Araki-Woods factors (Kunal Mukherjee of IIT, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar).

K-POTENT PRESERVING LINEAR MAPS

Noncommutative Hardy space associated with semi-finite subdiagonal algebras

Let ℳ be a von Neumann algebra with a faithful normal trace τ, and let H∞ be a finite, maximal. subdiagonal algebra of ℳ. We prove that the Hilbert transform associated with H∞ is a linear continuous map from L1 (ℳ, τ) into L1.∞ (ℳ, τ). This provides a non-commutative version of a classical theorem of Kolmogorov on weak type boundedness of the Hilbert transform. We also show that if a positive measurable operator b is such that b log+b ∈ L1 (ℳ, τ) then its conjugate b, relative to H∞ belongs to L1 (ℳ, τ). These results generalize classical facts from function algebra theory to a non-commutative setting.

The main objective of this workshop was to bring together experts from various fields, which are all interested in finite and finite-dimensional approximations of infinite algebraic and analytic objects, such as groups, algebras, dynamical systems, group actions, or even von Neumann algebras.

It is well-known that the algebraic tensor product E ⊗ Y of a not necessarily locally convex topological vector space E and a locally convex space Y can be identified with a subspace of the so-called e-product EeY (a space of continuous linear mappings from Y′ into E). So, whenever EeY is complete, even the completed tensor product is (isomorphic to) a subspace of EeY. As this occurs in many important cases, it is interesting to remark that, for each continuous linear operator u from a locally convex space F into E, there exists a locally convex U with continuous embedding j∶U→E and a continuous linear map u∶F→U such that u=j·u. As main applications of a combination of these ideas, we obtain a characterization of the functions in as continuous functions with values in locally convex spaces (this gives new aspects for the intergration theory of Gramsch [5]) and a result extending a theorem in [6] on holomorphic functions with values in non locally convex spaces to arbitrary complex manifolds.

The complete lattice generated by a totally atomic CSL ℒ and the projection lattice of a von Neumann algebra ℛ, commuting with ℒ, is reflexive. From this it follows that the strongly closed lattice generated by any CSL ℒ and the projection lattice of a properly infinite von Neumann algebra ℛ, commuting with ℒ, is reflexive.

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.

In this note we observe that the assertion in the title follows easily from known results. Recall that an algebra 6R of bounded linear operators on a Hilbert space is called [1] triangular if (R 6* is a maximal abelian von Neumann algebra. A triangular algebra is maximal if it is not properly contained in another triangular algebra; it is hyperreducible if the projections onto its invariant subspaces generate a maximal abelian von Neumann algebra. Now the theorem stated in the title can be proven as follows. Let a1 be a weakly closed maximal triangular algebra. Since aR is maximal the lattice of invariant subspaces of 61 is totally ordered [1, Lemma 2.3.3]. It follows from Theorem 2 of [2] that 61 is refexive, i.e., 61 contains every operator which leaves invariant every invariant subspace of (R. Hence, by Lemma 5.1 of [3], 61 is hyperreducible. This result shows one reason that hyperreducible maximal triangular algebras are nice (see [1]). Note that the proof of Lemma 5.1 of [3] shows that every reflexive triangular algebra (not just those with totally ordered invariant subspace lattices), is hyperreducible. Thus it follows (by a proof similar to the above) from Theorem 3 of [2] that a weakly closed triangular algebra whose lattice of invariant subspaces is the direct product of a family of totally ordered lattices is hyperreducible. In view of these results it seems reasonable to conjecture that every weakly closed triangular algebra is hyperreducible. A proof of this conjecture would be a partial answer to question (i) of [2].

We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extreme point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extreme, then the whole map is extreme. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.

Let $M_{i}$ and $N_{i}$ be von Neumann algebras and $\tau_{i}$ be completely positive maps from $M_{i}$ to $N_{i}(i=1,2)$ . Then there exists a completely positive map $\tau_{1}\otimes\tau_{2}$ (called the product map of $\tau_{1}$ and $\tau_{2}$ ) from the spatial $C^{*}$ -tensor product $M_{1}\otimes_{a}M_{2}$ to the spatial $C^{*}$ -tensor product $N_{1}\otimes_{\alpha}N_{2}$ such that $\tau_{1}\otimes\tau_{2}(a\otimes b)=\tau_{1}(a)\otimes\tau_{2}(b)$ .

The mixing properties of shift-automorphisms of an infinite tensor product of matrix algebras with a shift invariant (but, in general, non-product) state are studied. It is shown that if the centralizer of the von Neumann algebra is a type II1 factor, then the mixing properties of the shift automorphism correspond to the mixing properties of the underlying measure space shift.