Von Neumann Dimension Research Articles

Thin buildings

Let X be a building of uniform thickness q+1. L^2-Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L^2-cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q. The weighted cohomology makes sense for all real positive values of q, and is computed for small q. If the Davis complex of the Coxeter group is a manifold, a version of Poincare duality allows to deduce that the L^2-cohomology of a building with large thickness is concentrated in the top dimension.

The Strong Approximation Conjecture holds for amenable groups

Let G be a finitely generated group and G ▷ G 1 ▷ G 2 ▷ ⋯ be normal subgroups such that ⋂ k = 1 ∞ G k = { 1 } . Let A ∈ Mat d × d ( C G ) and A k ∈ Mat d × d ( C ( G / G k ) ) be the images of A under the maps induced by the epimorphisms G → G / G k . According to the strong form of the Approximation Conjecture of Lück [W. Lück, L 2 -Invariants: Theory and Applications to Geometry and K-theory, Ergeb. Math. Grenzgeb. (3), vol. 44, Springer-Verlag, Berlin, 2002] dim G ( ker A ) = lim k → ∞ dim G / G k ( ker A k ) , where dim G denotes the von Neumann dimension. In [J. Dodziuk, P. Linnell, V. Mathai, T. Schick, S. Yates, Approximating L 2 -invariants and the Atiyah conjecture, Comm. Pure Appl. Math. 56 (7) (2003) 839–873] Dodziuk et al. proved the conjecture for torsion free elementary amenable groups. In this paper we extend their result for all amenable groups, using the quasi-tilings of Ornstein and Weiss [D.S. Ornstein, B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987) 1–141].

Square Integrable Representations, von Neumann Algebras and An Application to Gabor Analysis

Let G be a locally compact group, and let Γ be a lattice in G. Let (π, Hπ ) be an irreducible, square integrable unitary representation of G. Then the restriction of π to Γ extends to VN(Γ), the von Neumann algebra generated by the regular representation of Γ. We determine the center valued (or extended) von Neumann dimension of Hπ as a VN(Γ)-module. This result is extended to representations which are square integrable modulo the center of G. An explicit formula is given when G is a semisimple algebraic group and when G is a nilpotent Lie group. As an application, we study the question of the existence of frames for the discretized windowed Fourier transform.

Amenable groups, topological entropy and Betti numbers

We investigate an analogue of theL 2-Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy.

Morse inequalities for covering manifolds

We study the existence of L2 holomorphic sections of invariant line bundles over Galois coverings. We show that the von Neumann dimension of the space of L2 holomorphic sections is bounded below under weak curvature conditions. We also give criteria for a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. As applications we prove the stability of the previous Moishezon pseudoconcave manifolds under perturbation of complex structures as well as weak Lefschetz theorems.

L2-index theorem for elliptic differential boundary problems

Suppose M is a compact manifold with boundary. Let N be a normal covering of M. Suppose (A,T) is an elliptic differential boundary value problem on M with lift (\tilde A,\tilde T) to N. Then the von Neumann dimension of kernel and cokernel of this lift are defined. The main result of this paper is: these numbers are finite, and their difference, by definition the von Neumann index, equals the index of (A,T). In this way, we extend the classical L^2-index theorem of Atiyah to manifolds with boundary.

The dimension monoid of a lattice

We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, $\Delta$ from L×L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Conc L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction: ¶¶ (1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to $\Bbb Z^+$ or to $\Bbb R^+$ .¶ (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power of ${\Bbb Z}^{+}\cup \{\infty\}.$ ¶ (3) If L is modular or if L has no infinite bounded chains, then Dim L is a refinement monoid.¶ (4) If L is a simple geometric lattice, then Dim L is isomorphic to $\Bbb Z^+$ , if L is modular, and to the two-element semilattice, otherwise.¶ (5) If L is an $\aleph_0$ -meet-continuous complemented modular lattice, then both Dim L and the dimension function $\Delta$ satisfy (countable) completeness properties.¶¶ If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M 2 (R), then Dim L is isomorphic to the monoid V (R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.

𝐿²-homology over traced *-algebras

Given a unital complex *-algebra A A , a tracial positive linear functional τ \tau on A A that factors through a *-representation of A A on Hilbert space, and an A A -module M M possessing a resolution by finitely generated projective A A -modules, we construct homology spaces H k ( A , τ , M ) H_k(A,\tau ,M) for k = 0 , 1 , … k = 0, 1, \ldots . Each is a Hilbert space equipped with a *-representation of A A , independent (up to unitary equivalence) of the given resolution of M M . A short exact sequence of A A -modules gives rise to a long weakly exact sequence of homology spaces. There is a Künneth formula for tensor products. The von Neumann dimension which is defined for A A -invariant subspaces of L 2 ( A , τ ) n L^2(A,\tau )^n gives well-behaved Betti numbers and an Euler characteristic for M M with respect to A A and τ \tau .

Bounds on the von Neumann dimension of $L\sp 2$-cohomology and the Gauss-Bonnet theorem for open manifolds

Bounds on the von Neumann dimension of $L\sp 2$-cohomology and the Gauss-Bonnet theorem for open manifolds