作用素環論の最近の進展 (Recent Developments in Operator Algebras)
RIMS ４２０, ２０１４年９月０８日(月)  １０日(水)

Monday, 08 
Tuesday, 09 
Wednesday, 10 
09:40  10:30 



10:40  11:30 





13:00  13:50 


Program in pdf 
14:00  14:50 




15:10  16:00 


16:10  17:00 


The classification of BaumslagSolitar groups up to measure equivalence is an open problem. In this talk, we consider free probability measure preserving actions of nonamenable BaumslagSolitar groups, whose canonical commensurated subgroup acts aperiodically. A new invariant for such actions allows us to cover the relative profinite completion of the BaumslagSolitar groups. This leads to a measure equivalence rigidity result for BaumslagSolitar groups. This is joint work with Cyril Houdayer.
Ozawa and Popa proved a unique factorization property for tensor products of free group factors. Roughly speaking, this means these tensor products "remember" each tensor component. In this seminar, we study similar factorization results for free quantum group factors. In the proof, we use a condition (AO) type phenomena for continuous cores of the tensor products, and prove first a weak factorization property on the cores. Then we deduce the desired property for the original tensor products.
In 1987, J. Rosenberg proved that if the reduced group $\mathrm{C}^*$algebra is quasidiagonal then the given group is amenable, and he conjectured that the converse also holds. We confirm this Rosenberg conjecture for elementary amenable groups. This is a joint work with N. Ozawa and M. Rørdam.
We discuss on the irreducible spherical unitary representations of Drinfeld doubles of $q$deformations of compact Lie groups and prove a partial classification result of that of $\mathrm{SU}_q(3)$.
This talk is based on joint work with JeanChristophe Bourin.
Let $\mathcal{M}$ be a finite von Neumann algebra with a faithful normal trace $\tau$, $\tau(1)=1$. A general exposition is given on the new notion of symmetric antinorms on $\mathcal{M}^+$ as well as symmetric norms on $\mathcal{M}$. We next consider special antinorms (called derived antinorms) $\A\_!:=\A^{p}\^{1/p}$ determined by a symmetric norm $\\cdot\$ on $\mathcal{M}$ and a constant $p>0$. For such antinorms and for a certain class of functions $\psi:[0,\infty)\to[0,\infty)$ we provide a general superadditivity inequality $$\tag{$\ast$} \\psi(A+B)\_!\ge\\psi(A)\_!+\\psi(B)\_!.$$ For $\alpha\in(0,1]$ consider the symmetric norm $$\X\_{(\alpha)}:=\int_0^\alpha\mu_t(X)\,dt,\qquad X\in\mathcal{M},$$ and the symmetric antinorm $$\Delta_\alpha(A):=\exp\biggl({1\over\alpha}\int_{1\alpha}^1\log\mu_t(A)\,dt\biggr), \qquad A\in\mathcal{M}^+,$$ where $\mu_t(X)$ denotes the generalized $s$number of $X\in\mathcal{M}$. Recall that the FugledeKadison determinant is $\Delta(X)=\Delta_1(X)$, $X\in\mathcal{M}$. From the limit formula $$\Delta_\alpha(A)=\lim_{p\searrow0}\alpha^{1/p}\A^{p}\_{(\alpha)}^{1/p}$$ for invertible $A\in\mathcal{M}^+$, we can apply $(\ast)$ to obtain $$\Delta_\alpha(\psi(A+B)) \ge\Delta_\alpha(\psi(A))+\Delta_\alpha(\psi(B)), \qquad A,B\in\mathcal{M}^+,\ \alpha\in(0,1], $$whose specialization to $\psi(t)=t$ and $\alpha=1$ is the Minkowski inequality of the FugledeKadison determinant $\Delta(A+B)\ge\Delta(A)+\Delta(B)$, $A,B\in\mathcal{M}^+$.
In 1977, Davidson proved that if an operator on the Hardy space commutes with all analytic Toeplitz operators modulo compact, then it must be a compact perturbation of a Toeplitz operator. We consider a generalization of Davidson's result to the semicrossed product of a von Neumann algebra, which is one of noncommutative generalizations of the classical analytic Toeplitz algebra.
QDを定義してから数年、AIとの関係を明らかにするべく努力しいるがいまだ達していない。ここでは流れに対する幾つかの概念を確認して、その間の微々たる進展を報告するとともに、いくつかの問題を述べてみたい。（QD $\mathrm{C}^*$環上の流れに対してはAIからQDが出る。QD流れとはある意味で行列環上の流れで近似できるもの。）
The Haagerup property for an arbitrary von Neumann algebra and its variants are introduced. I will present a sketchy proof of the equivalence of all these properties. Also, stability results are discussed.
This is the jointwork with R. Okayasu.
A couple of years ago I solved the questions of factoriality, type classification and fullness for arbitrary free product von Neumann algebras, and then clarified when a given type III$_1$ free product factor admits `discrete decomposition' by computing Connes's $\tau$invariant for arbitrary type III$_1$ free product factors. Probably, we can say that almost all basic invariants of free product von Neumann algebras become clear, and the next tasks should focus on much finer structure of free product factors under suitable assumptions. The talk indeed concerns one of my attempts in the direction.
In my recent work, it is proved that for every ICC group which is embeddable into a hyperbolic group (e.g. free group), the reduced group $\mathrm{C}^*$algebra is realizes as the intersection of a decreasing sequence of $\mathrm{C}^*$algebras all of which are isomorphic to the Cuntz algebra ${\mathcal O}_2$. To prove this, we construct amenable quotients of the boundary action.
arXiv:1406.2740
We construct and describe a new class of positive but not completely positive maps with random matrix tools. We show that these maps can be new entanglement witnesses in some cases.
This is a joint work with Ion Nechita.
Recently we have generalized "quantumclassical correspondence" for harmonic oscillators to the context of "interacting Fock spaces." Under a simple condition for Jacobi sequences, it is shown that the Arcsine law is the unique probability distribution corresponding to the large quantum number limits. As a corollary, we obtain that the squared nth orthogonal polynomials for a probability distribution corresponding to such kinds of interacting Fock spaces, multiplied by the probability distribution and normalized, weakly converge to the Arcsine law as n tends to infinity.
This is a joint work with Hayato Saigo.