Robert Lauter

• Johannes Gutenberg University Mainz

We establish a relation between two different approaches to a complete pseudodifferential analysis of totally characteristic or Fuchs type operators on compact manifolds with boundary respectively conical singularities: Melrose’s (overblown) b-calculus and Schulze’s cone algebra. Though quite different in their definition, we show that these two ps...

In (17, 19), Melrose has studied examples of non-compact mani- folds M0 whose large scale geometry is described by a Lie algebra of vector fields V ( M;TM) on a compactification of M0 to a manifold with corners M. The geometry of these manifolds - called "manifolds with a Lie struc- ture at infinity" - was studied from an axiomatic point of view in...

The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals th...

Several examples of non-compact manifolds $M_0$ whose geometry at infinity is described by Lie algebras of vector fields $V \subset \Gamma(TM)$ (on a compactification of $M_0$ to a manifold with corners $M$) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of...

The 0-calculus on a manifold with boundary is a micro-localization of the Lie algebra of vector fields that vanish at the boundary. It has been used by Mazzeo, Melrose to study the Laplacian of a conformally compact metric. We give a complete characterization of those 0-pseudodifferential operators that are Fredholm between appropriate weighted Sob...

The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals th...

We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contributi...

We construct and study several algebras of pseudodifferential operators that are closed under holomorphic functional calculus. This leads to a better understanding of the structure of inverses of elliptic pseudodifferential operators on certain non-compact manifolds. It also leads to decay properties for the solutions of these operators. To cite th...

We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, ``extended Weyl algebras,'' whose definition was inspired by Guillemin's paper on the subject. An extended Weyl algebra...

A manifold with a Lie structure at in nity" is a non-compact manifold M0 whose geometry is described by a compacti cation to a manifold with corners M and a Lie algebra of vector elds on M subject to constraints only on M r M0 . The Lie structure at in nity on M0 determines a metric on M0 up to bi-Lipschitz equivalence. This leads to the natural pr...

We recall a Fredholm criterion for fully elliptic cusp(pseudo)differential operators on a compact manifold with corners ofarbitrary codimension, acting on suitable Sobolev spaces. Then we give aformula for the index in terms of regularized `trace' functionalssimilar to the residue trace of Wodzicki and Guillemin.

A manifold with a ``Lie structure at infinity'' is a non-compact manifold $M_0$ whose geometry is described by a compactification to a manifold with corners M and a Lie algebra of vector fields on M, subject to constraints only on $M \smallsetminus M_0$. The Lie structure at infinity on $M_0$ determines a metric on $M_0$ up to bi-Lipschitz equivale...

We compute the Hochschild homology of the algebra of double-edge pseudo-differential operators. The double-edge algebra is naturally associated to a compact manifold whose boundary is the total space of a fibration of closed manifolds. We introduce residue-type traces on this algebra and on various of its ideals and quotients. As an application, we...

We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better underst...

We use a pseudodifferential calculus on differentiable groupoids to obtain new analytical results on geometric operators on certain noncompact Riemannian manifolds. The first step is to establish that the geometric op-erators belong to a pseudodifferential calculus on an associated differentiable groupoid. This then leads to Fredholmness criteria f...

We consider the calculus Ψ de(X, Ω) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects. We construct a scale of weighted double-edge Sobolev spaces on...

We use algebras of pseudodifferential operators on groupoids to study geometric operators on non-compact manifolds and singular spaces. The first step is to establish that the geometric operators are in our algebras. This then leads to criteria for Fredholmness for geometric operators on suitable non-compact manifolds, as well as to an inductive pr...

The author computes the Jacobson topology and the corresponding Borel structure on the spectrum of $$B(Z,{}^b\Omega ^{\tfrac{1}{2}} )$$ . Here $$B(Z,{}^b\Omega ^{\tfrac{1}{2}} )$$ denotes the C*-algebra generated by the algebra $$\Psi _{b,cl}^0 B(Z,{}^b\Omega ^{\tfrac{1}{2}} )$$ of b-pseudo-differential operators of orderO on a compact manifold wit...

We study properties and representations of the convo-lution algebra and the algebra of pseudodifferential operators asso-ciated to a continuous family groupoid. We show that the study of representations of the algebras of pseudodifferential operators of or-der zero completely reduces to the study of the representations of the ideal of regularizing...

We compute the length of the C*-algebra generated by the algebra of b-pseudodifferential operators of order 0 on compact manifolds with corners.

As a contribution to the pseudodifferential analysis on manifolds with singularities we construct for each smooth, compact manifold X with boundary a Ψ*-algebra ()∞(X, bΩ1/2)⊆(ϱL2(X, bΩ1/2)) containing the algebra Ψ0b, cl(X, bΩ1/2) of totally characteristic pseudodifferential operators introduced by Melrose [25] in 1981 as a dense subalgebra; furth...

Let X be a compact manifold with boundary. It will be shown (Theorem 3.4) that the small Melrose algebra A≔ ϕb,cl (χ,bΩ1/2) (cf. [22], [23]) of classical, totally characteristic pseudodifferential operators carries no topology such that it is a topological algebra with an open group of invertible elements, in particular, the algebra A cannot be spe...

Literaturverz. S. 230 - 239. Zugl.: Mainz, Univ., Diss., 1996.

If X is a Hilbert space, it is shown that very general subalgebras A of ℒ(X) contain the holomorphic functional calculus in several variables in the sense of J. L. Taylor. In particular, Taylor’s holomorphic functional calculus applies to Ψ * -algebras, and so gives a useful tool for the investigation of certain algebras of pseudo-differential oper...

Let X be a Banach space and A⊆ℒ(X) a Ψ 0 -algebra in ℒ(X) [B. Gramsch, Math. Ann. 269, No. 1, 27-71 (1984; Zbl 0661.47037), Definition 5.1]. Then for every commuting system a=(a 1 ,⋯,a n )∈A n an upper semicontinuous joint spectrum, the A-Taylor split spectrum σ Ts A (a,X) will be defined. Furthermore, for functions f analytic in a neighbourhood of...