Jerome Kaminker

Exactness and Uniform Embeddability of Discrete Groups

A numerical quasi-isometry invariant of a finitely generated group is defined whose values parametrize the difference between being uniformly embeddable in a Hilbert space and being exact.

Exactness and the Novikov conjecture

Abstract In this note we will study a connection between the conjecture that C ∗ r (Γ) is exact and the Novikov conjecture for Γ. The main result states that if the inclusion of the reduced C ∗ -algebra C r ∗ (Γ) of a discrete group Γ into the uniform Roe algebra of Γ, UC ∗ (Γ) , is a nuclear map then Γ is uniformly embeddable in a Hilbert space. By a result of Yu (Invent. Math. 139 (2000) 201) this implies that Γ satisfies the Novikov conjecture. Note that the hypothesis is a slight strengthening of the usual notion of exactness since a group Γ is exact if and only if the inclusion of C r ∗ (Γ) into B (l 2 (Γ)) is nuclear.

Cyclic cocycles, renormalization and eta-invariants

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K-Theoretic Duality for Shifts of Finite Type

Abstract:We will study the stable and unstable Ruelle algebras associated to a hyperbolic homeomorphism of a compact space. To do this, we will describe a notion of K-theoretic duality for -algebras which generalizes Spanier-Whitehead duality in topology. A criterion for checking that it holds is presented. As an application it is shown that the Ruelle algebras which are associated to subshifts of finite type, (and agree with Cuntz-Krieger algebras in this case) satisfy this criterion and hence are dual.

The spectrum of operators elliptic along the orbits of ℝ n actions

It is shown that a periodic elliptic operator on ℝn has no eigenvalues off of the set of discontinuities of its spectral density function. The methods involve operator algebras and are based on a “spectral duality” principal first introduced by J. Bellisard and D. Testard. A version of the spectral duality theorem is proved which relates the point spectrum of a certain family of operators to the continuous spectrum of an associated family.

The longitudinal cocycle and the index of Toeplitz operators

Abstract Let D be a self-adjoint leafwise elliptic operator on a foliated manifold. Compressing multiplication operators to the range of the positive spectral projection for D yields the class of leafwise Toeplitz operators. The extension generated by these operators is constructed. A topological formula for the index of a Toeplitz operator with invertible symbol is given. This index can also be obtained by pairing the K-theory class of the symbol with a certain cyclic cocycle. If one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle, then this cocycle can be used to obtain refined invariants of the original operator.

Geometric and Analytic Properties of Groups

We will present a survey of the connections between the harmonic analysis of a discrete group and the asymptotic properties of the group considered as a metric space.

Some Homotopy and Shape Calculations for C*-Algbbras

Suppose that a C*-algebra A is isomorphic to the direct limit \(\underrightarrow {\lim }\,\text{A}_\text{n} \) An of a system of C*-algebra and injections

C^*

\text{A}_\text{1} \xrightarrow{{\varphi _1 }}\text{A}_\text{2} \xrightarrow{{\varphi _2 }} \ldots \ldots