Sylvester matrix rank functions on crossed products

Abstract

In this paper we consider the algebraic crossed product${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$induced by a homeomorphism$T$on the Cantor set$X$, where$K$is an arbitrary field with involution and$C_{K}(X)$denotes the$K$-algebra of locally constant$K$-valued functions on$X$. We investigate the possible Sylvester matrix rank functions that one can construct on${\mathcal{A}}$by means of full ergodic$T$-invariant probability measures$\unicode[STIX]{x1D707}$on$X$. To do so, we present a general construction of an approximating sequence of$\ast$-subalgebras${\mathcal{A}}_{n}$which are embeddable into a (possibly infinite) product of matrix algebras over$K$. This enables us to obtain a specific embedding of the whole$\ast$-algebra${\mathcal{A}}$into${\mathcal{M}}_{K}$, the well-known von Neumann continuous factor over$K$, thus obtaining a Sylvester matrix rank function on${\mathcal{A}}$by restricting the unique one defined on${\mathcal{M}}_{K}$. This process gives a way to obtain a Sylvester matrix rank function on${\mathcal{A}}$, unique with respect to a certain compatibility property concerning the measure$\unicode[STIX]{x1D707}$, namely that the rank of a characteristic function of a clopen subset$U\subseteq X$must equal the measure of$U$.

$$L^2$$-Betti numbers arising from the lamplighter group

“…This measure is ergodic, full and T -invariant, so we can apply the techniques developed in [2] to study the Z-crossed product algebra A := C K (X ) T Z by giving 'μapproximations' of the space X , which at the level of the algebra A correspond to certain 'approximating' * -subalgebras A n ⊆ A (see [2,Section 4.1], also [3,Section 6]). By using [2, Theorem 4.7 and Proposition 4.8], we obtain a canonical faithful Sylvester matrix rank function rk A on A which coincides, in case K is a subfield of C closed under complex conjugation, with the rank function rk K Γ on the group algebra K Γ naturally inherited from the canonical rank function in the * -regular ring U(Γ ) [3,Proposition 5.10].…”

“…In light of this, one can define 'generalized' 2 -Betti numbers in this more general setting, that is, arising from the Z-crossed product algebra A = C K (X ) T Z, for K an arbitrary field and T an arbitrary homeomorphism on a Cantor set X . Turning back to the lamplighter group algebra K Γ , Graboswki has shown in a recent paper [18] the existence of irrational (in fact transcendental) 2 -Betti numbers arising from Γ , exhibiting a concrete example in [18,Theorem 2]. With this result, the lamplighter group has become the simplest known example which gives rise to irrational 2 -Betti numbers.…”

“…Our main result is the following: Theorem 1.1 (Theorem 4.5) Let Γ be the lamplighter group. Then for any family of natural numbers {d i | 1 ≤ i ≤ n}, and any family of polynomials { p i (x) | 0 ≤ i ≤ n} of positive degrees and nonnegative integer coefficients, such that the linear coefficient of p 0 (x) is nonzero, there exists a matrix operator A over QΓ such that b (2)…”

“…This theory provides a well-defined notion of dimension for H D , called the von Neumann dimension of H D . This new framework led Atiyah to define the so-called 2 -Betti numbers as von Neumann dimensions of 2 -cohomology Hilbert modules [33].…”

“…Although they were defined analytically, 2 -Betti numbers can be defined in an algebraic setting, purely in terms of the group G and without explicit mention of the manifold M (see Sect. 2.2 for a detailed explanation).…”