I followed Chapter one's brief window into techniques and properties matrix amplification and elementary dilation results, and then proceeded to introduce operator spaces and positive maps. Paulsen pp. 1-9.
Dilation Theory Learning Seminar
Details
The University of Haifa is hosting a learning seminar in Dilation Theory. Our key text will be Paulsen's Completely Bounded Maps and Operator Algebras, before including additional resources. We will each read from the text leading up to our weekly meetings, and a volunteer from among us will present on the relevant section. There is a Discord server available where participants may answer and ask questions leading up to the seminar-- please email Lucas Hall at the address below to receive an invitation link. Beginning the week of January 14 2024, we will be meeting in-person on Tuesdays from 5:30-7:00 PM IST in room #814 in the Amado building at the Technion university campus. We will continue to broadcast the seminar virtually for those of you unable to attend physically.
Requirements
This seminar is designed to minimize prerequisites, so that it will be accessible to as broad an audience as possible. We will assume some familiarity with the structure of a C*-algebra, particularly developments up to and including the GNS-representation.
Upcoming
Past
November 28, 2023. Lucas Hall (Haifa) Introduction to Dilation Theory
December 5, 2023. Prahllad Deb (IIIT Delhi) Positive Maps
We investigate the norms of positive maps, and conditions under which a linear map is positive. Paulsen pp. 9-13.
December 12, 2023. Lucas Hall (Haifa) Techniques for Building Positive Maps and Consequences
A parametrization of positive maps by contractions (Theorem 2.6), related consequences, and other techniques to build positive maps. Comments on the nonunital setting. Operator spaces and n-positive maps. Paulsen pp. 13-18; 26-27. (omit Spectral Sets).
January 1, 2024. Zuly Salinas (Technion) Completely Positive Maps
n-positivity and the completely adverb. Elementary Properties and complete positivity for commutative C*-algebras. Paulsen pp. 26-34.
January 8, 2024. Lucas Hall (University of Haifa) Further Results on CP Maps. Applications to Operator Theory.
Steinspring's Completely Positive Theorem and Choi's Theorem. Consequences for Operator Theory, and module mappings. Paulsen pp. 32-39.
January 16, 2024. Lucas Hall (Haifa) Dilation Theorems
Steinspring's Dilation Theorem, and minimal Steinspring representations.
January 23, 2024. Boris Bilich (Haifa University) Further Dilation Theorems
Sz. Nagy's Dilation Theorem, Operator valued measures, Naimark's Dilation Theorem.
January 30, 2024. Pavel Prudnikov (University of Haifa) Complete positivity for matrix algebras and group maps.
Completely positive maps between matrix algebras. Results for maps from groups.
February 6, 2024. Sibaprasad Barik (Technion) Commuting Isometries
The dilation of commuting isometries to commuting untitaries, dilations of certain isometric semigroup homomorphisms to unitary representations. Ando's Dilation Theorem, the two-variable von Neumann inequality. Paulsen p. 58-63
February 20, 2024. Jeet Sampat (Technion) Completely Positive Maps into Matrix Algebras
The relationship between maps into matrix algebras and linear functionals for C*-algebras, operator systems, and operator spaces. Some applications. Paulsen p. 73-80
March 5, 2024. Lucas Hall (U Haifa) Arveson's Extension Theorems
Topological preliminaries for Banach Spaces leading to the extension theorems. The definition of an operator algebra and the study of their extensions.
March 12, 2024. Boris Bilich (UHaifa) Maximal Dilations and the U.E.P.
Characteristics of dilations for u.c.p. maps. Maximality and the Unique Extension Property.
March 19, 2024. Pavel Prudnikov (Technion) The C*-envelope and Boundary Representations.
The Dritschel-McCullough Theorem and the existence of the C*-envelope.
Contact
email: hallluc1 at msu dot edu
Office: Wells Hall C320
Michigan State University
Department of Mathematics
619 Red Cedar Rd.
East Lansing, MI 48924