Journal of Topology and Analysis

Let F be a non archimedean local field and let G be an algebraic connected almost F-simple group over F, whose Lie algebra contains sl₃(F). We prove that G(F) has strong Banach property (T) in a stronger sense than in the article "Un renforcement de la propriété (T)", published in Duke Math. J. As a consequence, families of expanders built from a lattice in G(F) do not embed uniformly in Banach spaces of type > 1. Also any affine isometric action of G(F) on a Banach space of type > 1 has a fixed point.

For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have "geometric K-theory", namely the "transmission algebra" introduced by Boutet de Monvel [5], the "zero algebra" introduced by Mazzeo in [9, 10] and the "scattering algebra" from [16], we give explicit formulas for the Chern character of the index bundle in terms of the symbols (including normal operators at the boundary) of a Fredholm family of fiber operators. This involves appropriate descriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bundle over a manifold with boundary in which the Chern character, mapping from the corresponding realization of K-theory, naturally takes values.

Answering a question of G. Yu, we show that the group EL₃(ℤ 〈 x, y 〉) is not linear over any field. We also prove that for every unitary associative ring R, the group EL_k(R) (k ≥ 3) is linear over ℂ if and only if R has a finite index ideal that is linear over ℂ.

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let A_ζ denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_ζ, the group of unitaries of A_ζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

For a tuple A = (A₁, A₂, …, A_n) of elements in a unital algebra over ℂ, its projective spectrumP(A) or p(A) is the collection of z ∈ ℂⁿ, or respectively z ∈ ℙ^n-1 such that A(z) = z₁A₁ + z₂A₂ + ⋯ + z_nA_n is not invertible in . In finite dimensional case, projective spectrum is a projective hypersurface. When A is commuting, P(A) looks like a bundle over the Taylor spectrum of A. In the case is reflexive or is a C*-algebra, the projective resolvent setP^c(A) := ℂⁿ \ P(A) is shown to be a disjoint union of domains of holomorphy. -valued 1-form A^-1(z)dA(z) reveals the topology of P^c(A), and a Chern–Weil type homomorphism from invariant multilinear functionals to the de Rham cohomology is established.