On a locally compact group we introduce covariant quantization schemes and analogs of phase space representations as well as mixed-state localization operators. These generalize corresponding notions ...for the affine group and the Heisenberg group. The approach is based on associating to a square integrable representation of the locally compact group two types of convolutions between integrable functions and trace-class operators. In the case of non-unimodular groups these convolutions only are well-defined for admissible operators, which is an extension of the notion of admissible wavelets as has been pointed out recently in the case of the affine group.
We introduce the notion of city products of right-angled buildings that produces a new right-angled building out of smaller ones. More precisely, if M is a right-angled Coxeter diagram of rank n and ...Δ1,…,Δn are right-angled buildings, then we construct a new right-angled building ▪. We can recover the buildings Δ1,…,Δn as residues of Δ, but we can also construct a skeletal building of type M from Δ that captures the large-scale geometry of Δ.
We then proceed to study universal groups for city products of right-angled buildings, and we show that the universal group of Δ can be expressed in terms of the universal groups for the buildings Δ1,…,Δn and the structure of M. As an application, we show the existence of many examples of pairs of different buildings of the same type that admit (topologically) isomorphic universal groups, thereby vastly generalizing a recent example by Lara Beßmann.
In this paper we discuss the Lp-Lq boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups G for the range 1<p≤2≤q<∞. As a consequence of the ...established Fourier multiplier theorem we also derive a spectral multiplier theorem on general locally compact separable unimodular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the Lp-Lq norms of the heat kernels for general positive unbounded invariant operators on G. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group, as well as for higher order operators. We show that our results imply the known results for Lp-Lq multipliers such as Hörmander's Fourier multiplier theorem on Rn or known results for Fourier multipliers on compact Lie groups. The new approach developed in this paper relies on advancing the analysis in the group von Neumann algebra and its application to the derivation of the desired multiplier theorems.
Let ω be a weight function defined on a locally compact group G, 1⩽p<+∞, S⊂G and let us assume that for any s∈S, the left translation operator Ts is continuous from the weighted Lp-space Lp(G,ω) into ...itself. For a given set Γ⊂C, a vector f∈Lp(G,ω) is said to be (Γ,S)-dense if the set {λTsf:λ∈Γ,s∈S} is dense in Lp(G,ω). In this paper, we characterize the existence of (Γ,S)-dense vectors in Lp(G,ω) in terms of the weight and the set Γ.
We give a detailed description of all totally disconnected locally compact groups with only compact elements in which any two closed subgroups are permutable and their product being a closed subgroup ...again. This amounts to sharpening the classification results in 2, Section 14 while not being dependent upon them.
In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings R, in particular for R=Z and R=Q. We show these properties satisfy many ...analogous results to the case of discrete groups, and we provide analogues of the famous Bieri's and Brown's criteria for finiteness properties and deduce that both FPn-properties and Fn-properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness properties.
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