Convolution-Dominated Operators on Discrete Groups

Abstract

Consider a discrete group G and a bounded self-adjoint convolu-tion operator T on l 2 (G); let σ(T) be the spectrum of T. The spectral theorem gives a unitary isomorphism U between l 2 (G) and a direct sum n L 2 (∆n, ν), where ∆n ⊂ σ(T), and ν is a regular Borel measure supported on σ(T). Through this isomorphism T corresponds to multiplication by the identity function on each summand. We prove that a nonzero function f ∈ l 2 (G) and its transform Uf cannot be simultaneously concentrated on sets V ⊂ G, W ⊂ σ(T) such that ν(W) and the cardinality of V are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

Left inverses of matrices with polynomial decay

“…The proof is completely similar to that of Theorem 1.7 using the fact, proved in [9] (see also [24] for a weaker statement) that C ω (G) is a spectral involutive algebra for all admissible weight.…”

Truncation Approximations and Spectral Invariant Subalgebras in Uniform Roe Algebras of Discrete Groups

“…Furthermore, by using the same technical lemma of Bickel and Lindner [2], we are able to show (by a quite short proof) that, for a group with subexponential growth, the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra (Theorem 4.9). This result can be viewed as a generalization of a recent result by Fendler, Gröchenig and Leinert [9] which shows that the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra for groups with polynomial growth.…”

“…Hence W is a dense Banach subalgebra of C * u (G) via left convolutions on 2 (G). In [9], Fendler, Gröchenig, Leinert showed that if G is amenable and rigidly symmetric, then W is a spectral invariant subalgebra of B( 2 (G)). The main examples of amenable and rigidly symmetric groups are groups with polynomial growth.…”

Convolution dominated operators on compact extensions of abelian groups

“…Proof. Like in [7] we define a representation R of L, but this time on L 2 (G) = i∈H L 2 (U). The image of R will turn out to be CD H .…”

“…Since H is amenable and l ∞ (H, B(L 2 (U))) is a C * algebra, by Leptin [20] the Mregular representation is a maximal representation of B. It is weakly equivalent to R. To see this one slightly modifies the proof of [7,Prop 3]. Since R happens on L 2 (G) = l 2 (H, L 2 (U)) and λ M happens on l 2 (H, L 2 (G)) = l 2 (H × H, L 2 (U)), we work on this last space, which we denote H for short.…”