Basic elements of modular theory

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

With the foundational principles having been put in place by the previous three chapters, we are now able to present a fairly comprehensive theory of tracial noncommutative Lp and Orlicz spaces corresponding to semifinite von Neumann algebra, which closely mimics the classical theory. In the tracial setting the theory of decreasing rearrangements provides a very close connection between noncommutative Lp and Orlicz spaces and their classical commutative cousins. We start the analysis by first proving the basic structural theory for noncommutative Lp-spaces (including trace duality), before introducing the reader to Orlicz spaces. For Orlicz spaces trace duality is replaced by Köthe duality. In closing we discuss the spaces L1, L∞, and L1+L∞ showing that they may respectively be interpreted as smallest and largest Orlicz spaces.

The Banach-Stone theorem says, in the language we shall use, that if two commutative C*-algebras are isomorphic, then they are as C*-algebras. In this formulation there are two natural question: whether the theorem holds for noncommutative C*-algebras, and whether it is possible to replace isometrically isomorphic with isomorphic. A complete answer to the first question was given by Kadison in [10], and M. Cambern settled the second in the affirmative in [5]. In Section 3 we show that Cambern's result may be partially extended to noncommutative von Neumann algebras having the property P ([9, 18, 19]). When we have the result of Section 3 that a nearly isometric completely positive map of a von Neumann algebra having property P is close to a * isomorphism between the algebras, we have nearly proved that von Neumann algebras whose unit balls are close in the Hausdorff metric are unitarily equivalent by a unitary close to the identity. The only thing we have to prove then is that if q) is an isomorphism of a v o n Neumann algebra A having property P onto a v o n Neumann algebra B acting on the same Hilbert space, and 4~ satisfies, for some positive k less than one and any x in A,

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such thatfor all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, thenwhere U is a bounded invertible operator on X and A′ is the adjoint of A.

Perturbations of certain crossed product algebras by free groups

Given a locally compact quantum group $\mathbb{G}$ and two $\mathbb{G}$-$W^{*}$-algebras $\alpha : A\curvearrowleft \mathbb{G}$ and $\beta : B\curvearrowleft \mathbb{G}$, we study the notion of equivariant $W^{*}$-Morita equivalence $(A, \alpha )\sim _{\mathbb{G}} (B, \beta )$, which is an equivariant version of Rieffel’s notion of $W^{*}$-Morita equivalence. We prove that important dynamical properties of $\mathbb{G}$-$W^{*}$-algebras, such as (inner) amenability, are preserved under equivariant Morita equivalence. For a coideal von Neumann algebra $L^\infty (\mathbb{K}\backslash \mathbb{G})\subseteq L^\infty (\mathbb{G})$ with dual coideal von Neumann algebra $L^\infty (\check{\mathbb{K}})\subseteq L^\infty (\check{\mathbb{G}})$, we use a natural $\check{\mathbb{G}}$-$W^{*}$-Morita equivalence $L^\infty (\mathbb{K}\backslash \mathbb{G})\rtimes _\Delta \mathbb{G} \sim _{\check{\mathbb{G}}} L^\infty (\check{\mathbb{K}})$ to relate dynamical properties of $L^\infty (\mathbb{K}\backslash \mathbb{G})$ with dynamical properties of $L^\infty (\check{\mathbb{K}})$. We use this to refine some recent results established by Anderson-Sackaney and Khosravi. This refinement allows us to answer a question of Kalantar, Kasprzak, Skalski, and Vergnioux, namely that for $\mathbb{H}$ a closed quantum subgroup of the compact quantum group $\mathbb{G}$, coamenability of $\mathbb{H}\backslash \mathbb{G}$ and relative amenability of $\ell ^\infty (\check{\mathbb{H}})$ in $\ell ^\infty (\check{\mathbb{G}})$ are equivalent. Moreover, if $\mathbb{G}$ is compact, we study the relation between $\mathbb{G}$-$W^{*}$-Morita equivalence of $(A, \alpha )$ and $(B, \beta )$ and $\mathbb{G}$-$C^{*}$-Morita equivalence of the associated $\mathbb{G}$-$C^{*}$-algebras $(\mathcal{R}(A), \alpha )$ and $(\mathcal{R}(B), \beta )$ of regular elements.

On square roots of the Haar state on compact quantum groups

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

The usual definition of (non-contextual) hidden variables is found to be too restrictive, in the sense that, according to it, even some classical systems do not admit hidden variables. A more general concept is introduced and the term “approximate hidden variables” is used for it. This new concept avoids the aforementioned problems, since all classical systems admit approximate hidden variables. Standard quantum systems do not admit approximate hidden variables, unless the corresponding Hilbert space is 2-dimensional. However, an appropriate non-standard quantum system, which arises by focussing on momentum and position and neglecting the remaining observables, admits approximate hidden variables. Systems associated with JBW-algebras (resp. von Neumann algebras) and satisfying some mild conditions admit approximate hidden variables iff they are classical, that is, iff the corresponding JBW-algebra (resp. von Neumann algebra) is associative (resp. commutative).

In [18], the author initiated an investigation of compact, Banach-module-valued derivations of C*-algebras. In collaboration with C. A. Akemann [3] and S.-K. Tsui [16], he determined the structure of all compact and weakly compact, A-valued derivations of a C*-algebra A, and of all compact, B(H)-valued derivations of a C*-subalgebra of B(H), the algebra of bounded linear operators on a Hilbert space H. In this paper, we begin the study of weakly compact, B(H)-valued derivations of C*-subalgebras of B(H).Let R be a C*-subalgebra of B(H), δ:R → B(H) a weakly compact derivation, i.e., a weakly compact linear map which hasSince δ has a unique weakly compact extension to a derivation of the closure of R in the weak operator topology (WOT) on B(H) (consult the proof of Theorem 3.1 of [16]), we may assume with no loss of generality that R is a von Neumann subalgebra of B(H). In this paper, we determine in Lemma 4.1 and Theorems 4.3 and 4.10 the structure of δ when R is type I, using I. E. Segal's multiplicity theory [14] for type I von Neumann algebras and results of E. Christensen [6], [7] on B(H)-valued derivations of von Neumann algebras.

We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum, and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum.

A quasi-expectation of a C ∗ {C^ \ast } -algebra A on a C ∗ {C^\ast } -subalgebra B is a bounded linear projection Q : A → B Q:A \to B such that Q ( x a y ) = x Q ( a ) y ∀ x , y ∈ B , a ∈ A Q(xay) = xQ(a)y \forall x,y \in B,a \in A . It is shown that if M is a von Neumann algebra of operators on Hilbert space H for which there exists a quasi-expectation of B ( H ) B(H) on M, then there exists a projection of norm one of B ( H ) B(H) on M, i.e. M is injective. Further, if M is an amenable von Neumann subalgebra of a von Neumann algebra N, then there exists a quasi-expectation of N on M ′ ∩ N M’ \cap N . These two facts yield as an immediate corollary the recent result of A. Connes that all amenable von Neumann algebras are injective.

Smooth points in von Neumann algebras are characterized in terms of minimal projections. The theorem generalizes known results for the algebra L ∞ ( Ω , Σ , μ ) {L^\infty }(\Omega ,\Sigma ,\mu ) and the space of bounded linear operators on a Hilbert space.

We use a non-commutative generalization of the Banach Principle to show that the classical individual ergodic theorem for subsequences generated by means of uniform sequences can be extended to the von Neumann algebra setting.

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).