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Register a new userThe Fuglede-Kadison determinant, theme and variations
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Authors
Pierre de la Harpe
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We review the definition of determinants for finite von Neumann algebras, due to Fuglede and Kadison (1952), and a generalisation for appropriate groups of invertible elements in Banach algebras, from a paper by Skandalis and the author (1984). After some reminder on K-theory and Whitehead torsion, we hint at the relevance of these determinants to the study of $L^2$-torsion in topology.
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We present a new approach to the Marcinkiewicz interpolation inequality for the distribution function of the Hilbert transform, and prove an abstract version of this inequality. The approach uses logarithmic determinants and new estimates of canonical products of genus one.
Grothendieck and Harder proved that every principal bundle over the projective line with split reductive structure group (and trivial over the generic point) can be reduced to a maximal torus. Furthermore, this reduction is unique modulo automorphisms and the Weyl group. In a series of six variations on this theme, we prove corresponding results for principal bundles over the following schemes and stacks: (1) a line modulo the group of nth roots of unity; (2) a football, that is, an orbifold of genus zero with two marked points; (3) a gerbe over a football whose structure group is the nth roots of unity; (4) a chain of lines meeting in nodes; (5) a line modulo an action of a split torus; and (6) a chain modulo an action of a split torus. We also prove that the automorphism groups of such bundles are smooth, affine, and connected.
Schinzel and Wojcik have shown that if $alpha, beta$ are rational numbers not $0$ or $pm 1$, then $mathrm{ord}_p(alpha)=mathrm{ord}_p(beta)$ for infinitely many primes $p$, where $mathrm{ord}_p(cdot)$ denotes the order in $mathbb{F}_p^{times}$. We begin by asking: When are there infinitely many primes $p$ with $mathrm{ord}_p(alpha) > mathrm{ord}_p(beta)$? We write down several families of pairs $alpha,beta$ for which we can prove this to be the case. In particular, we show this happens for 100% of pairs $A,2$, as $A$ runs through the positive integers. We end on a different note, proving a version of Schinzel and W{o}jciks theorem for the integers of an imaginary quadratic field $K$: If $alpha, beta in mathcal{O}_K$ are nonzero and neither is a root of unity, then there are infinitely many maximal ideals $P$ of $mathcal{O}_K$ for which $mathrm{ord}_P(alpha) = mathrm{ord}_P(beta)$.
Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalisation structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.
Let $A$ be a unital $C^*$-algebra and let $U_0(A)$ be the group of unitaries of $A$ which are path connected to the identity. Denote by $CU(A)$ the closure of the commutator subgroup of $U_0(A).$ Let $i_A^{(1, n)}colon U_0(A)/CU(A)rightarrow U_0(mathrm M_n(A))/CU(mathrm M_n(A))$ be the hm, defined by sending $u$ to ${rm diag}(u,1_n).$ We study the problem when the map $i_A^{(1,n)}$ is an isomorphism for all $n.$ We show that it is always surjective and is injective when $A$ has stable rank one. It is also injective when $A$ is a unital $C^*$-algebra of real rank zero, or $A$ has no tracial state. We prove that the map is an isomorphism when $A$ is the Villadsens simple AH--algebra of stable rank $k>1.$ We also prove that the map is an isomorphism for all Blackadars unital projectionless separable simple $C^*$-algebras. Let $A=mathrm M_n(C(X)),$ where $X$ is any compact metric space. It is noted that the map $i_A^{(1, n)}$ is an isomorphism for all $n.$ As a consequence, the map $i_A^{(1, n)}$ is always an isomorphism for any unital $C^*$-algebra $A$ that is an inductive limit of finite direct sum of $C^*$-algebras of the form $mathrm M_n(C(X))$ as above. Nevertheless we show that there are unital $C^*$-algebras $A$ such that $i_A^{(1,2)}$ is not an isomorphism.
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