September 0509 KTGU Lecture 
Sci 3127 
Sorin Popa (UCLA/Kyoto)
Paving over arbitrary MASAs in von Neumann algebras
05(M) 15:00  17:00, 06(T) 15:00  17:00, 07(W) 15:00  17:00, 08(T) 15:00  17:00, 09(F) 15:00  17:00.
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) $D$ of $B(H)$ extends to a unique state on $B(H)$. They also showed that this unique pure state extension property is equivalent to norm paving over $D$ for operators in $B(H)$. The KadisonSinger paving problem has been recently solved in the affirmative by MarcusSpielmanSrivastava. In these lectures, we will introduce a general paving property for a MASA $A$ in a von Neumann factor $M$, called sopaving, involving approximation in the sotopology, rather than in norm, but which coincides with normpaving in the case $D\subset B(H)$. We conjecture that sopaving 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.
