�ì—p‘fŠÂ˜_‚̉Ä�‡�h (Summer Camp on Operator Algebras)
�¬’M, 2014”N07ŒŽ19“ú - 22“ú Otaru (Hokkaido), July 19 - 22, 2014

Organizers: R. Okayasu, N. Ozawa (chief), and R. Tomatsu

Cyril Houdayer: Structure of free product von Neumann algebras.

I will survey various recent results regarding the structure of free products of arbitrary von Neumann algebras.
Lecture 1: I will review several tools that are useful in the structure theory of von Neumann algebras. These include Cartan subalgebras, Popa's intertwining techniques, Connes-Takesaki's continuous decomposition and ultraproduct von Neumann algebras.
Lecture 2: I will outline the proof of a joint result with R. Boutonnet and S. Raum showing that free products of arbitrary von Neumann algebras have no Cartan subalgebra.
Lecture 3: I will outline the proof of a recent result showing that any diffuse amenable von Neumann algebra is maximal amenable inside its free product with an arbitrary von Neumann algebra.

Narutaka Ozawa�FNoncommutative Real Algebraic Geometry.

I will give an introduction to the emerging (?) subject of "noncommutative real algebraic geometry," a subject which deals with equations and inequalities in noncommutative algebras over the reals, with the help of analytic tools such as representation theory and operator algebras.

Takahiro Hasebe: Free independence and free cumulants

Free independence is the abstraction of relationship of generators of a free group. I would like to explain free cumulants which give us clear understanding of free independence.

Yuhei Suzuki: Some isomorphism theorems of O_2 (Theorem of Elliott and Rørdam)

The goal of this minilecture is to give a proof of Elliott--Rørdam's isomorphism theorem: \bigotimes _{n\in N}O_2 \cong O_2. The study of the endomorphism on the UHF-core of O_2 and that of homomorphisms on O_2 are the key of the proof.