Fredholm Properties of Band-dominated Operators on Periodic Discrete Structures

Abstract

Let (X, ∼) be a combinatorial graph the vertex set X of which is a discrete metric space. We suppose that a discrete group G acts freely on (X, ∼) and that the fundamental domain with respect to the action of G contains only a finite set of points. A graph with these properties is called periodic with respect to the group G. We examine the Fredholm property and the essential spectrum of band-dominated operators acting on the spaces l p (X) or c0(X), where (X, ∼) is a periodic graph. Our approach is based on the thorough use of band-dominated operators. It generalizes the necessary and sufficient results obtained in [39] in the special case X = G = Z n and in [42] in case X = G is a general finitely generated discrete group. Mathematics Subject Classification (2000). Primary 47B36; Secondary 47B39, 47A53.

Spatial discretization of restricted group C^*-algebras

“…The following theorem is due to Roe [22], see also [11]. Recall in this connection that a group Γ is said to be exact, if its reduced translation algebra is an exact C * -algebra.…”

“…Note that this result holds as well if the left regular representation is replaced by the right regular one and if, thus, the operators L s and R t change their roles. In fact, in [11,22] the results are presented in this symmetric setting. In [11] we showed moreover that the uniform boundedness condition in Theorem 4.7 is redundant for band operators if the group Γ has sub-exponential growth and if not every element of Γ is cyclic in the sense that w n = e for some positive integer n. For details see [11].…”

“…In fact, in [11,22] the results are presented in this symmetric setting. In [11] we showed moreover that the uniform boundedness condition in Theorem 4.7 is redundant for band operators if the group Γ has sub-exponential growth and if not every element of Γ is cyclic in the sense that w n = e for some positive integer n. For details see [11]. Note that the condition of sub-exponential growth is satisfied by the abelian groups Z N , the discrete Heisenberg group and, more general, by nilpotent groups (in fact, these groups have polynomial growth), whereas the growth of the free groups F N is exponential.…”

Arveson Dichotomy and Essential Fractality

“…These methods have found fruitful applications in the stability analysis of different approximation methods for numerous classes of operators; see the monographs [7,8,17] for an overview. In particular, I would like to emphasize the finite sections method for band-dominated operators, a topic which was mainly influenced and shaped by Vladimir S. Rabinovich and the limit operator techniques developed by him, see [9,10,11,12,13] and [15] for an overview. In fact, the algebra of the finite sections method for band-dominated operators is the first real-life example of an essentially fractal, but not fractal, algebra (these notions will be introduced below).…”

Multiplication operators on the weighted Lipschitz space of a tree

“…Operators on discrete structures other than the Laplacian have been studied in a number of papers (e.g., see the works of Pavone [11], [12], [13], Roe [14], and Rabinovich and Roch [15], [16], and [17]). Examples include the composition operators on L p spaces associated with homogeneous trees, the Toeplitz operators on discrete groups, and the band-dominated operators defined on p (X), where X is a discrete metric space.…”