# Operator algebras on manifolds: from the boundaryless to the boundary case

#### 24.11.2021, 14:00  –  Golm, Haus 9, 2.22 und Zoom Institutskolloquium

Elmar Schrohe (Leibniz University Hannover) und Ryszard Nest (University of Copenhagen)

2 pm: Elmar Schrohe (Leibniz University Hannover): Introduction to a Theorem by Duistermaat and Singer

3:15 pm: Ryszard Nest (University of Copenhagen): Order Preserving Isomorphisms of Boutet de Monvel's Algebra

Elmar Schrohe (Leibniz University Hannover): Introduction to a Theorem by Duistermaat and Singer

Abstract: In a remarkable work, Duistermaat and Singer in 1976 studied the algebras of all classical pseudodifferential operators on smooth (boundaryless) manifolds. They gave a description of order preserving algebra isomorphism between the algebras of classical pseudodifferential operators of two manifolds under a cohomological assumption pertaining the first manifold. `Order preserving' here means that the isomorphism preserves the order of the operators. Surprisingly no continuity assumption is necessary; continuity is automatic. Mathai and Melrose later showed in 2017 that the cohomological assumption is not needed, at least if the manifolds are compact.

The first talk serves as an introduction to the topic. We will explain the notion of Fourier integral operator and the reason for their appearance in Duistermaat and Singer's description of the order preserving algebra isomorphisms. In joint work in progress we want to determine the order-preserving isomorphisms in the case of manifolds with boundary. We expect them to be given by conjugation by the Fourier integral operators on manifolds with boundary developed in joint work with U. Battisti and S. Coriasco. The operator algebras involved here are Boutet de Monvel algebras which we shall introduce, describing their specific features.

Ryszard Nest (University of Copenhagen): Order Preserving Isomorphisms of Boutet de Monvel's Algebra

In the second talk we will explain the specific features appearing in the construction of the Fourier integral operator implementing the isomorphism as well as some very simple geometrical, analytic and cohomological constructions that are involved. The main fact of life about manifold with boundary is that vector fields do not define global flows and the "boundary conditions" are a way of dealing with this problem. The Boutet de Monvel algebra corresponds to the choice of local boundary conditions and is, effectively, a non-commutative completion of the manifold. One can think of it as a parametrised version of the classical Toeplitz algebra as a completion of the half-space. Once this is explained, the analysis that we need reduces to a high degree to relatively classical results about automorphisms and homology of the Toeplitz algebra and we will explain what those are and how they are used.