Published Jan 02, 2024
Introduction to operator spaces and completely bounded maps.
A student must be very comfortable with techniques of functional analysis, commensurate with PMath 450/650 at U Waterloo. I will also assume a basic comfort level with C*-algebras: positivity, GNS construction, representability on Hilbert spaces; von Neumann algebras: double commutation ; commensurate, for example with PMath 810 or any topics course in operator algebras at U Waterloo.
| Understand the need for, and work with operator space structures and completely bounded maps. |
Rough outline
Projective tensor product of Banach spaces and duality.
Operator spaces, completely bounded maps; operator systems, completely positive maps: Stinespring’s theorem, Arveson’s and Wittstock's extension theorems.
Abstract operator spaces & systems, Ruan's representation theorem.
New operator spaces from old: quotient spaces, mapping & dual spaces, row and column Hilbert spaces.
Tensor products: projective & injective operator space tensor products, duality; Haagerup tensor product, multiplicative multilinearity, abstract operator algebras.
Optional topics: approximation properties and some tensor products of C*-algebras, injective envelopes, interpolation theory, tensor products of operator systems.
| Title / Name | Notes / Comments | Required |
|---|---|---|
| Operator Spaces, Effros and Ruan | Main reference | No |
| Completely Bounded Maps and Operator Algebras, Paulsen | Main reference | No |
| C*-algebras and Finite-Dimensional Approximations, Brown and Ozawa | Secondary reference | No |
| Introduction to Operator Space Theory, Pisier | Secondary reference | No |
Lectures will be self contained, up to prerequisite expectations, so no book is necessary to purchase.
| Component | Value |
|---|---|
| Assignments (3-4) | 85% |
| Talk (topics will be made available after Reading Week) | 15% |
I will accept assignments by paper, or by e-mail if typed.
Talks will (most likely) be scheduled after last day of class.
No assignment screening will be used in this course.
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Note for students with disabilities: AccessAbility Services, located in Needles Hall, Room 1401, collaborates with all academic departments to arrange appropriate accommodations for students with disabilities without compromising the academic integrity of the curriculum. If you require academic accommodations to lessen the impact of your disability, please register with AccessAbility Services at the beginning of each academic term.
Turnitin.com: Text matching software (Turnitin®) may be used to screen assignments in this course. Turnitin® is used to verify that all materials and sources in assignments are documented. Students' submissions are stored on a U.S. server, therefore students must be given an alternative (e.g., scaffolded assignment or annotated bibliography), if they are concerned about their privacy and/or security. Students will be given due notice, in the first week of the term and/or at the time assignment details are provided, about arrangements and alternatives for the use of Turnitin in this course.
It is the responsibility of the student to notify the instructor if they, in the first week of term or at the time assignment details are provided, wish to submit alternate assignment.