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17 - Paving over arbitrary MASAs in von Neumann algebras, 2016

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Super Global Course Special Lectures 1

Paving over arbitrary MASAs in von Neumann algebras

Sorin Popa
Kyoto University / Distinguished Visiting Professor
University of California, Los Angeles / Professor

Sep. 5, 6, 7, 8 and 9, 2016
Room127, Graduate School of Science Bldg No 3

Lecture Video

Course Description

Motivated by an intriguing claim in Dirac's 1947 book on "Quantum Mechanics", Kadison and Singer have asked the question of whether any pure state on the diagonal maximal abelian subalgebra (MASA) 01.png of 02.png extends to a unique state on 02.png . They also showed that this unique pure state extension property is equivalent to norm paving over 01.png for operators in 02.png. The Kadison--Singer paving problem has been recently solved in the affirmative by Marcus--Spielman--Srivastava.

In these lectures, we will introduce a general paving property for a MASA 03.png in a von Neumann factor 04.png , called so-paving, involving approximation in the so-topology, rather than in norm, but which coincides with norm-paving in the case 05.png . We conjecture that so-paving holds true for any MASA in any factor. We check the conjecture in many cases, including singular and regular MASAs in hyperfine factors. Related problems will be discussed.