Von Neumann Dimension Research Articles

On Bergman space zero sets. Editors’ comments on the article by Vaughan Jones

This article has two purposes. The first one is to describe what the Editors know on how and when Vaughan Jones worked on the subject of the article published in the same volume. The starting point, in the late 1980’s, was his fascination for a formula giving Murray–von Neumann dimensions of Hilbert spaces of unitary representations of Fuchsian groups. Over the years, he discovered surprising relations of these dimensions with other domains of mathematics. The second purpose is to expose with some details a subject which plays an important role in Jones’ article: the irreducible projective unitary representations of \operatorname{PSL}_2(\mathbf{R}) , which constitute a continuous family known as the discrete series, and which have interesting restrictions to various discrete subgroups of \operatorname{PSL}_2(\mathbf{R}) .

Bergman space zero sets, modular forms, von Neumann algebras and ordered groups

A^2_{\alpha} will denote the weighted L^2 Bergman space. Given a subset S of the open unit disc we define \Omega(S) to be the infimum of \{s \mid \exists f \in A^2_{s-2}, \; f\neq 0, having S as its zero set \} . By classical results on Hardy space there are sets S for which \Omega(S)=1 . Using von Neumann dimension techniques and cusp forms we give examples of S where 1<\Omega(S)<\infty . By using a left order on certain Fuchsian groups we are able to calculate \Omega(S) exactly if \Omega (S) is the orbit of a Fuchsian group. This technique also allows us to derive in a new way well known results on zeros of cusp forms and indeed calculate the whole algebra of modular forms for \operatorname{PLS}_2 (\mathbb{Z}) .

The product of lattice covolume and discrete series formal dimension: [formula omitted]-adic [formula omitted

Let F be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in PGL(2,F) and the formal dimension of a discrete series representation of GL(2,F). The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.

Affine density, von Neumann dimension and a problem of Perelomov

We provide a solution to Perelomov's 1972 problem concerning the existence of a phase transition (known in signal analysis as ‘Nyquist rate’) determining the basis properties of certain affine coherent states labelled by Fuchsian groups. As suggested by Perelomov, the transition is given according to the hyperbolic volume of the fundamental region. The solution is a more general form (in phase space) of the PSL(2,R) variant of a 1989 conjecture of Kristian Seip about wavelet frames, where the same value of ‘Nyquist rate’ is obtained as the trace of a certain localization operator. The proof consists of first connecting the problem to the theory of von Neumann algebras, by introducing a new class of projective representations of PSL(2,R) acting on non-analytic Bergman-type spaces. We then adapt to this setting a new method for computing von Neumann dimensions, due to Sir Vaughan Jones. Our solution contains necessary conditions in the form of a ‘Nyquist rate’ dividing frames from Riesz sequences of coherent states and sampling from interpolating sequences. They hold for an infinite sequence of spaces of polyanalytic functions containing the eigenspaces of the Maass operator and their orthogonal sums. Within mild boundaries, we show that our result is best possible, by characterizing our sequence of function spaces as the only invariant spaces under the non-analytic PSL(2,R)-representations.

The density theorem for projective representations via twisted group von Neumann algebras

We consider converses to the density theorem for square-integrable, irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that the restriction of such a σ-projective unitary representation π of a unimodular, second-countable group G to a lattice Γ extends to a Hilbert module over the twisted group von Neumann algebra of (Γ,σ). We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-cocycle satisfying Kleppner's condition, we show that the center-valued von Neumann dimension reduces to the scalar value dπvol(G/Γ), where dπ is the formal dimension of π and vol(G/Γ) is the covolume of Γ in G. We apply our results to characterize the existence of multiwindow super frames and Riesz sequences associated to π and Γ. In particular, we characterize when a lattice in the time-frequency plane of a second-countable, locally compact abelian group admits a Gabor frame or Gabor Riesz sequence.

Canonical measures on metric graphs and a Kazhdan’s theorem

We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of "hyperbolic measures" on universal covers of metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. We will prove a generalized version of this theorem for metric graphs, allowing any infinite Galois cover to replace the universal cover. We will show all such limiting measures satisfy a version of Gauss-Bonnet formula which, using the theory of von Neumann dimensions, can be interpreted as a "trace formula". In the special case where the infinite cover is the universal cover, we will provide explicit methods to compute the corresponding limiting (hyperbolic) measure. Our ideas are motivated by non-Archimedean analytic and tropical geometry.

The Realization Problem for some wild monoids and the Atiyah Problem

The Realization Problem for (von Neumann) regular rings asks what are the conical refinement monoids which can be obtained as the monoids of isomorphism classes of finitely generated projective modules over a regular ring. The analogous realization question for the larger class of exchange rings is also of interest. A refinement monoid is said to be wild if it cannot be expressed as a direct limit of finitely generated refinement monoids. In this paper, we consider the problem of realizing some concrete wild refinement monoids by regular rings and by exchange rings. The most interesting monoid we consider is the monoid M \mathcal M obtained by successive refinements of the identity x 0 + y 0 = x 0 + z 0 x_0+y_0=x_0+z_0 . This monoid is known to be realizable by the algebra A = K [ F ] A= K[\mathcal F] of the monogenic free inverse monoid F \mathcal F , for any choice of field K K , but A A is not an exchange ring. We show that, for any uncountable field K K , M \mathcal M is not realizable by a regular K K -algebra, but that a suitable universal localization Σ − 1 A \Sigma ^{-1}A of A A provides an exchange, non-regular, K K -algebra realizing M \mathcal M . For any countable field F F , we show that a skew version of the above construction gives a regular F F -algebra realizing M \mathcal M . Finally, we develop some connections with the Atiyah Problem for the lamplighter group. We prove that the algebra A A can be naturally seen as a ∗ * -subalgebra of the group algebra k G kG over the lamplighter group G = Z 2 ≀ Z G= \mathbb {Z}_2\wr \mathbb {Z} , for any subfield k k of C \mathbb {C} closed under conjugation, and we determine the structure of the ∗ * -regular closure of A A in U ( G ) \mathcal U(G) . Using this, we show that the subgroup of R \mathbb R generated by the von Neumann dimensions of matrices over k G kG contains Q \mathbb {Q} .

On the growth of von Neumann dimension of harmonic spaces of semipositive line bundles over covering manifolds

We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action, and obtain an asymptotic estimate for the von Neumann dimension of the space of harmonic [Formula: see text]-forms with values in high tensor powers of a semipositive line bundle. In particular, we estimate the von Neumann dimension of the corresponding reduced [Formula: see text]-Dolbeault cohomology group. The main tool is a local estimate of the pointwise norm of harmonic forms with values in semipositive line bundles over Hermitian manifolds.

Further properties of [formula omitted] dimension

This article establishes more properties of the ℓp dimension introduced in a previous article. Given an amenable group Γ acting by translation on ℓp(Γ), a number satisfying dimension-like properties is associated to the (usually infinite dimensional) subspaces Y of ℓp(Γ) which are invariant under the action of Γ. This may be interpreted as a von Neumann dimension for ℓp spaces. As a consequence, for p∈[1,2], if Y is a closed non-trivial Γ-invariant subspace of ℓp(Γ) and let Yn be an increasing sequence of closed Γ-invariant subspace such that ⋃Yn¯=ℓp(Γ;V), then there exist a k such that Yk∩Y≠{0}.

An [formula omitted]-version of von Neumann dimension for Banach space representations of sofic groups

In [5], A. Gournay defined a notion of lp-dimension for Γ-invariant subspaces of lq(Γ)⊕n, with Γ amenable. The number dimlqlp(Γ,V) is dimV when p=q, and is preserved by a certain class of Γ-equivariant bounded linear isomorphisms. We develop a notion of dimlp,Σ(Y,Γ) where Y is a Banach space with a uniformly bounded action of a sofic group Γ and Σ is a sofic approximation. In particular, our definition makes sense for a large class of non-amenable groups. We also develop a notion of dimSp,Σ(Y,Γ) with Γ an Rω-embeddable group and Sp the space of finite-dimensional Schatten p-class operators. These numbers are invariant under bounded Γ-equivariant linear isomorphisms and under the natural translation action of Γ, dimlp(lp(Γ,V),Γ)=dimV, and dimSp(lp(Γ,V),Γ)=dimV for 1⩽p⩽2. In particular, this shows that lp(Γ,V) is not isomorphic to lp(Γ,W) as a representation of Γ if dimV≠dimW, and Γ is Rω-embeddable. We discuss other concrete computations in a follow-up paper, including proving that our dimension agrees with von Neumann dimension for representations contained in a multiple of the left-regular representation.

Rational group ring elements with kernels having irrational dimension

We prove that there are examples of finitely generated groups G together with group ring elements Q \in \bbQ G for which the von Neumann dimension \dim_{LG}\ker Q is irrational, so (in conjunction with other known results) answering a question of Atiyah.

Free lamplighter groups and a question of Atiyah

We compute the von Neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational, thus providing an elementary constructive answer to a question of Atiyah.

Applications of Følner's condition to quantum groups

Using the Følner condition for coamenable quantum groups we derive information about the ring-theoretical structure of the Hopf algebras arising from such quantum groups, as well as an approximation result concerning the Murray–von Neumann dimension associated with the corresponding the von Neumann algebra.

A Planar Calculus for Infinite Index Subfactors

We develop an analog of Jones' planar calculus for II_1-factor bimodules with arbitrary left and right von Neumann dimension. We generalize to bimodules Burns' results on rotations and extremality for infinite index subfactors. These results are obtained without Jones' basic construction and the resulting Jones projections.

The Atiyah Conjecture and Artinian Rings

Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N(G), and let D(KG,U(G)) denote the division closure of KG in U(G); thus D(KG,U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, and \alpha \in \Mat_n(KG). Then \alpha induces a bounded linear map \alpha: l^2(G)^n \to \l^2(G)^n, and \ker\alpha has a well-defined von Neumann dimension \dim_{N(G)} (\ker\alpha). This is a nonnegative real number, and one version of the Atiyah conjecture states that d \dim_{N(G)}(\ker\alpha) \in Z. Assuming this conjecture, we shall prove that if G has no nontrivial finite normal subgroup, then D(KG,U(G)) is a d \times d matrix ring over a skew field. We shall also consider the case when G has a nontrivial finite normal subgroup, and other subrings of U(G) that contain KG.

Formula omitted] dimensions of spaces of braid-invariant harmonic forms

Let X be a Riemannian manifold endowed with a co-compact isometric action of an infinite discrete group. We consider L2 spaces of harmonic vector-valued forms on the product manifold XN that are invariant with respect to an action of the braid group BN, and compute their von Neumann dimensions (the braided L2-Betti numbers).

Error-Correcting Codes and Phase Transitions

The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of this paper is to relate the asymptotic bound to phase diagrams of quantum statistical mechanical systems. We first identify the code parameters with Hausdorff and von Neumann dimensions, by considering fractals consisting of infinite sequences of code words. We then construct operator algebras associated to individual codes. These are Toeplitz algebras with a time evolution for which the KMS state at critical temperature gives the Hausdorff measure on the corresponding fractal. We extend this construction to algebras associated to limit points of codes, with non-uniform multi-fractal measures, and to tensor products over varying parameters.

A dynamical approach to von Neumann dimension

Let $\Gamma$ be an amenable group and $V$ be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of <i>l</i>$^2(\Gamma;V)$ (with respect to $\Gamma$) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a $\Gamma$-invariant linear subspaces $Y$ of <i>l</i>$^p(\Gamma;V)$ a real positive number dim<SUB><i>lp</i></SUB> Y (which is the von Neumann dimension when $p=2$). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective $\Gamma$-equivariant linear map of finite-type from <i>l</i>$^p(\Gamma;V) \to $<i>l</i>$^p(\Gamma; V')$ if $\dim V > \dim V'$. A generalization of the Ornstein-Weiss lemma is developed along the way.

Pseudo-Rank Functions on Rickart *-rings

Abstract. Pseudo-rank functions on Rickart *-rings are introduced and their propertiesare studied. 1. IntroductionA real valued function Don a lattice Lis called a dimension function if therange of D has either an upper bound or a lower bound and for all a;b 2L,D(a_b) + D(a^b) = D(a) + D(b), see; von Neumann [12] p.58. The theory ofdimension functions is studied in various structures. von Neumann [12] introduceddimensionality in continuous geometries by using perspectivity, whereas Iwamura[6] used the concept of a relation called the p-relation .Kaplansky [8], Murray and von Neumann [11] and others have introduced dimen-sionality in rings of operators by using equivalence of projections. Maeda [10] gener-alized the work of von Neumann [12] and Kaplansky [8] for a certain class of lattices.At the same time Loomis [9] gave an abstract setting to the Murray, von Neumanndimension theory by using complete orthocomplemented lattices. Berberian [2] hasdeveloped theory of dimension functions on the lattice of projections of a nite Baer*-ring. Goodearl [4] developed the dimension theory for a certain class of modules.von Neumann [12], p.231 has introduced the concept of a rank-function on a regularring which generalizes the dimension function. Goodearl [3], [5] has introduced anddeveloped the study of pseudo-rank functions on regular rings, which is a general-ization of rank functions.In this paper we introduce and study the concept of a pseudo-rank function on aRickart *-ring R. We obtain some basic properties of pseudo-rank functions andthe set of all pseudo-rank functions on R, on the lines of Goodearl [5] for Rickart*-rings. The unde ned terms are from Berberian [2] and Birkho [1].

WeightedL2–cohomology of Coxeter groups

Given a Coxeter system $(W,S)$ and a positive real multiparameter $\bq$, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $\Sigma$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W,S)$ and the multiparameter $q$. They have a "von Neumann dimension" with respect to the associated "Hecke - von Neumann algebra," $N_q$. The dimension of the $i^th$ cohomology group is denoted $b^i_q(\Sigma)$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b^i_q(\Sigma)$ are the usual $L^2$-Betti numbers of buildings of type $(W,S)$ and thickness $q$. For a certain range of $q$, we calculate these cohomology groups as modules over $N_q$ and obtain explicit formulas for the $b^i_q(\Sigma)$. The range of $q$ for which our calculations are valid depends on the region of convergence of the growth series of $W$. Within this range, we also prove a Decomposition Theorem for $N_q$, analogous to a theorem of L. Solomon on the decomposition of the group algebra of a finite Coxeter group.