Relative Yoneda Cohomology for Operator Spaces

The commutant lifting theorem of [24] may be regarded as a very general interpolation theorem from which a number of classical interpolation results may be deduced. In this paper we prove a spectral version of the commutant lifting theorem in which one bounds the spectral radius of the interpolant and not the norm. We relate this to a spectral analogue of classical matricial Nevanlinna-Pick interpolation.

This chapter contains the proofs of the operator-valued interpolation theorems introduced in the previous chapter. The proofs are based on the commutant lifting theorem which is stated for convenience in the first section. In Chapter IV the commutant lifting theorem will be revisited in more detail, and a constructive proof of this theorem will be given there.

<p>The classical Nevanlinna-Pick interpolation theorem, proved in 1915 by Pick and in 1919 by Nevanlinna, gives a condition for when there exists an interpolating function in H∞(D) for a specified set of data in the complex plane. In 1967, Sarason proved his commutant lifting theorem for H∞(D), from which an operator theoretic proof of the classical Nevanlinna-Pick theorem followed. Several competing noncommutative generalizations arose as a consequence of Sarason's result, and two strategies emerged for proving generalized Nevanlinna-Pick theorems: via a commutant lifting theorem or via a resolvent, or displacement, equation.</p><p>We explore the difference between these two approaches. Specifically, we compare two theorems: one by Constantinescu-Johnson from 2003 and one by Muhly-Solel from 2004. Muhly-Solel's theorem is stated in the highly general context of W*-correspondences and is proved via commutant lifting. Constantinescu-Johnson's theorem, while stated in a less general context, has the advantage of an elegant proof via a displacement equation. In order to make the comparison, we first generalize Constantinescu-Johnson's theorem to the setting of W*-correspondences in Theorem 3.0.1. Our proof, modeled after Constantinescu-Johnson's, hinges on a modified version of their displacement equation. Then we show that Theorem 3.0.1 is fundamentally different from Muhly-Solel's. More specifically, interpolation in the sense of Muhly-Solel's theorem implies interpolation in the sense of Theorem 3.0.1, but the converse is not true. Nevertheless, we identify a commutativity assumption under which the two theorems yield the same result.</p><p>In addition to the two main theorems, we include smaller results that clarify the connections between the notation, space of interpolating maps, and point evaluation employed by Constantinescu-Johnson and those employed by Muhly-Solel. We conclude with an investigation of the relationship between Theorem 3.0.1 and Popescu's generalized Nevanlinna-Pick theorem proved in 2003.</p>

The fifteenth Appalachian Set Theory workshop was held at Carnegie Mellon University in Pittsburgh on March 3, 2012. The lecturer was Asger Tornquist. As a graduate student Martino Lupini assisted in writing this chapter, which is based on the workshop lectures. Introduction The aim of the lectures is to give a brief introduction to the area of von Neumann algebras to a typical set theorist. The ideal intended reader is a person in the field of (descriptive) set theory, who works with group actions and equivalence relations, and who is familiar with the rudiments of ergodic theory, and perhaps also orbit equivalence. This should not intimidate readers with a different background: Most notions we use in these notes will be defined. The reader is assumed to know a small amount of functional analysis. For those who feel a need to brush up on this, we recommend consulting [Ped89]. What is the motivation for giving these lectures, you ask. The answer is two-fold: On the one hand, there is a strong connection between (non-singular) group actions, countable Borel equivalence relations and von Neumann algebras, as we will see in Lecture 3 below. In the past decade, the knowledge about this connection has exploded, in large part due to the work of Sorin Popa and his many collaborators. Von Neumann algebraic techniques have lead to many discoveries that are also of significance for the actions and equivalence relations themselves, for instance, of new cocycle superrigidity theorems.

In this note, we prove a harmonic-type maximal principle for the Schur parametrization of all intertwining liftings of an intertwining contraction in the commutant lifting theorem.

Sorin Popa [Po 1] [Po 2] has proved several results on the existence and properties of hyperfinite subfactors R of a type II1 factor M with the relative commutant R0 \M of R in M equal to C1. These theorems have been used in various cohomology calculations [CES, CS, PS, CPSS], as averaging over an amenable subgroup that generates the hyperfinite subfactor is a major step in showing that the continuous and completely bounded Hochschild cohomology groups are equal. It has seemed reasonable that Popa's results could be extended from factors to general type II1 von Neumann algebras by direct integral theory [KR, Chapter 4]. However, we do not know of such an attempt. Direct integral theory can be used directly to prove cohomology is zero and deduce results like [CPSS, Theorems 5.4 and 5.5] however these theorems on the continuous Hochschild cohomology for a von Neumann algebra with Cartan subalgebras are deduced from the theorems in this paper. This paper provides direct proofs of Popa's main two results in [Po 1] by modifying his proofs using an interpolation type result for projections in a maximal abelian selfadjoint sub-algebra (masa) of the type II1 algebra. This introduction contains a more detailed description of how our results extend Popa's, the basic definitions, and a brief reference to their use in the calculation of Hochschild cohomology groups in von Neumann algebras. Though averaging plays an important role in calculating the Hochschild cohomology of von Neumann algebras for all types I ; II1; II1; III of von Neumann algebras (see [Ri]), the results proved here are only used in the type II1 situation. The reason is that the type I 's are already trivially hyperfinite, and the type II1 and III von Neumann algebras may be handled by their stability under tensoring with B H†. This tensor factor B H† of M in the II1 and III cases gives a suitable hyperfinite algebra over which to average. Popa [Po 1] restricts his attention to type II II1 and II1† factors. MATH. SCAND. 85 (1999), 105^120

A dual Banach algebra is a Banach algebra that is also a dual Banach space such that multiplication is separately weak\(^*\) continuous. Examples of dual Banach algebras are, among others, von Neumann algebras, the measure algebra M(G). and the Fourier–Stieltjes algebra B(G) of a locally compact group G, or the algebras \(\mathcal {B}(E)\) of all bounded linear operators on a reflexive Banach space E.

K-POTENT PRESERVING LINEAR MAPS

Geometry of C⁎-algebras, and the bidual of their projective tensor product

Let A \mathfrak {A} be a UHF algebra and A ( D ) \mathcal {A}({\mathbf {D}}) the disk algebra. If A = [ ∪ n ≥ 1 A n ] − \mathfrak {A} = {\left [ {{ \cup _{n \geq 1}}{\mathfrak {A}_n}} \right ]^ - } and α \alpha is a product-type automorphism of A \mathfrak {A} which leaves each A n {\mathfrak {A}_n} invariant, then α \alpha defines an embedding \[ A n ⊗ A ( D ) ↪ ı n A n + 1 ⊗ A ( D ) \mathfrak {A}_n \otimes \mathcal {A}({\mathbf {D}}) \stackrel {\imath _n}{\hookrightarrow } {\mathfrak {A}_{n + 1}} \otimes \mathcal {A}({\mathbf {D}}) \] . The inductive limit of this system is a Banach algebra whose maximal ideal space is closely related to that of the disk algebra if the Connes spectrum Γ ( α ) \Gamma (\alpha ) is finite.

Nonlinearity inH ?-control theory, causality in the commutant lifting theorem, and extension of intertwining operators

The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

In this note we give another proof of the commutant lifting theorem (see) [6] and [7]), based on the Adamjan-Arov-Krein techniques introduced in [1]. We then apply the construction given in Theorem 1 below to obtain a generalization of a result in [2] (see also [4]).

In this note a commutant lifting theorem for vector-valued functional Hilbert spaces over generalized analytic polyhedra in $\Bbb{C}^n$ is proved. Let $T$ be the compression of the multiplication tuple $M_z$ to a $*$-invariant closed subspace of the under

This paper is concerned with two notions of cohomological triviality for Banach algebras, weak amenability and cyclic amenability. The first is defined within Hochschild cohomology and the latter within cyclic cohomology. Our main result is that where ℱ is a Banach algebraic free product of two Banach algebras and ℬ. It follows that cyclic amenability is preserved under the formation of free products.