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

Conditional expectations in von Neumann algebras and a theorem of Takesaki

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,

Given a strongly continuous orthogonal representation (U_t)_{t\in \mathbb{R}} of \mathbb{R} on a real Hilbert space {\mathcal{H}}_{\mathbb{R}} , a decomposition {\mathcal{H}}_{\mathbb{R}}:=\bigoplus_{i\in N}\mathcal{H}_{\mathbb{R}}^{(i)} consisting of invariant subspaces of (U_t)_{t\in\mathbb{R}} and an appropriate matrix ((q_{ij}))_{N\times N} of real parameters, we associate representations of the mixed commutation relations on twisted Fock spaces. The associated von Neumann algebras are (usually) non-tracial and are generalizations of those constructed by Bożejko–Speicher and Hiai. We investigate the factoriality of these von Neumann algebras. Along the process, we show that the generating abelian subalgebras associated with the blocks of the aforesaid decomposition are strongly mixing masas when they admit appropriate conditional expectations. On the contrary, the generating abelian algebras which fail to admit appropriate conditional expectations are quasi-split. We also discuss non-injectivity and the Haagerup approximation property.

To be able to comprehend and study noncommutative Lp and Orlicz spaces for general possibly non-semifinite von Neumann algebras, some background in modular theory and crossed products is essential. Starting with a review of the GNS-construction we give a brief overview of the essentials of modular theory. We then present various extensions of the Radon-Nikodym theorem proved in Chapter 3 including the enigmatic Connes cocycle derivatives, after which we review the existence criteria for faithful normal conditional expectation in the general setting. Next we briefly describe how modular theory may also be approached from a Hilbert space rather than operator algebraic perspective. Here the natural positive cone plays a crucial role. In conclusion we take the trouble to also give a thumbnail introduction to group von Neumann algebras for locally compact groups boasting amongst other features a new proof of the existence of the Plancherel weight and culminating in the definition of locally compact Quantum Groups.

1. Introduction. The conditional expectation has been studied by several authors, $e$ . $g$ . $[1]$ F. Combes, [5] I. Kov\'acs and J. Sz\"uces,

Let [Formula: see text] be a von Neumann algebra with a finite trace τ, represented in [Formula: see text], and let [Formula: see text] be sub-algebras, for t in an interval I (0 ∈ I). Let [Formula: see text] be the unique τ-preserving conditional expectation. We say that the path t ↦ Et is smooth if for every [Formula: see text] and [Formula: see text], the map [Formula: see text] is continuously differentiable. This condition implies the existence of the derivative operator [Formula: see text] If this operator satisfies the additional boundedness condition, [Formula: see text] for any closed bounded subinterval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras [Formula: see text] are *-isomorphic. More precisely, there exists a curve [Formula: see text], t ∈ I of unital, *-preserving linear isomorphisms which intertwine the expectations, [Formula: see text] The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps [Formula: see text] onto [Formula: see text]. We show that this restriction is a multiplicative isomorphism.

Let A be a function algebra, X its Silov boundary, M its maximal ideal space. One of the outstanding problems in function theory is to find a structure of complex analytic space on M such that the functions of A are analytic for this analytic structure. In some cases (algebras with few representative measures, or polynomial hulls of regular curves in ~n) , such structures can be described. (See also the recent work of R. Basener [I] and of Gamelin, Sibony and Wermer, this conference.) The first counterexample to the existence of analytic structure was found by Stolzenberg [14] who exhibited a compact K c ~2 such that the polynomial hull h(K) of K is distinct from K , and there exis~ no analytic structure on h(K) K . More explicit examples were found by Wermer and Basener for rationally comvex hulls (see [2] and also [6] for related class of examples). Moreover, if an analytic structure exists, it would have to live on the Gleason parts of the maximal ideal space. As a consequence,if f is a non constant function in a Gleason part P , then f(P) would have to be interior point to f(M), by the fact that analytic functions are open mappings.~nd that assertion is completely false in general. The interest for the research of analytic structure is that it provides regularity theorems on the functions of A , and also theorems about representative measures, bounded point derivation, approximation theory, etc. But such theorems, in particular differentiability theorems about the functions of A , can be proved if one can find in the maximal ideal space, some kind of dynamical system along the trajectories of which one can study the properties of functions. The aim of this paper is to show how such generalized dynamical systems (brownian paths or even sheets) lives in spectrum of certain function algebras even if there is no analytic structure in these spectrumnand to obtain from that regularity theorems, theorems about Jensen measures, approximation and derivations, just as if there was analytic structure.

The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra \mbox{\cal A} on the groupoid \Gamma = E \times G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of \mbox{\cal A}, is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein's equation. The algebra \mbox{\cal A}, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra \mathcal{M} of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which depends on the state \phi . The same state defines the noncommutative probability measure (in the sense of Voiculescu's free probability theory). Moreover, the state \phi satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra \mbox{\cal A}, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not ``feel'' singularities.

We prove that the additive group (E*, τ k (E)) of an $${\mathcal {L}_\infty}$$ -Banach space E, with the topology τ k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ k (E)) is uniformly homeomorphic to a subset of $${\ell_2^\kappa}$$ for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.

We show that a real Banach algebra $A$ such that $\Vert a^{2}\Vert =\Vert a\Vert^{2}$ for $a\in A$ is a subalgebra of the algebra $C_{\mathbb{H}}( X) $ of continuous quaternion-valued functions on a compact set $X$.

Finite subalgebras of a von Neumann algebra

Let (M, ϕ) = (M1, ϕ1) * (M2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M1 ⊂ M in arbitrary free product von Neumann algebras.

Almost sure convergence in noncommutative L2-spaces.- Individual ergodic theorems in L2 over a von neumann algebra.- Asymptotic formulae.- Convergence of iterates of contractions.- Convergence of orthogonal series and strong laws of large numbers.- Convergence of conditional expectations and martingales.- Miscellaneous results.

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).