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Narutaka Ozawa

Publications and source records attributed to Narutaka Ozawa.

At least 55 records · Page 3Linked to original sources
arXiv2010

On Ulam stability

We study $ε$-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional $ε$-represendations are uniformly close to unitary representations. One of our main results is that certain lattices in connected semi-simple Lie groups of higher rank are Ulam stable. For infinite-dimensional $ε$-representations, the similarly defined notion of strong Ulam stability is defined and it is shown that groups with free subgroups are not strongly Ulam stable. We also study deformation rigidity of unitary representations and show that groups containing a free subgroup are not deformation rigid.

math.FA↗
arXiv2010

Quasi-homomorphism rigidity with noncommutative targets

As a strengthening of Kazhdan's property (T) for locally compact groups, property (TT) was introduced by Burger and Monod. In this paper, we add more rigidity and introduce property (TTT). This property is suited for the study of rigidity phenomena for quasi-homomorphisms with noncommutative targets. Partially upgrading a result of Burger and Monod, we will prove that SL(n,R) with n at least 3 and their lattices have property (TTT). As a corollary, we generalize the well-known fact that every homomorphism from such a lattice into an amenable group or a hyperbolic group has finite image to the extent that it includes a quasi-homomorphism.

math.GR↗
arXiv2009

The Dixmier problem, lamplighters and Burnside groups

J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier.

math.GR↗
arXiv2008

On a class of II_1 factors with at most one Cartan subalgebra II

This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan subalgebras. We also prove a rigidity result for some group measure space II_1 factors.

math.OA↗
arXiv2008

An example of a solid von Neumann algebra

We prove that the group-measure-space von Neumann algebra $L^\infty(T^2) \rtimes SL(2,Z)$ is solid. The proof uses topological amenability of the action of $SL(2,Z)$ on the Higson corona of $Z^2$.

math.OA↗
arXiv2007

Weak Amenability of Hyperbolic Groups

We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combinatorial approach in the spirit of Haagerup and prove that for the word length metric d on a hyperbolic group, the Schur multipliers associated with r^d have uniformly bounded norms for 0<r<1. We then combine this with a Bozejko-Picardello type inequality to obtain weak amenability.

math.FA↗
arXiv2007

A Kurosh type theorem for type II_1 factors

We prove a Kurosh type theorem for free-product type II_1 factors. In particular, if M = LF_2 \otimes R, then the free-product type II_1 factors M*...*M are all prime and pairwise non-isomorphic. This paper is a continuation of [N. Ozawa, Solid von Neumann algebras] and [N. Ozawa and S. Popa, Some prime factorization results for type II_1 factors].

math.OA↗
arXiv2007

Boundaries of Reduced Free Group C*-Algebras

We prove that the crossed product $C^*$-algebra $C^*_r(Γ,\partialΓ)$ of a free group $Γ$ with its boundary $\partialΓ$ naturally sits between the reduced group $C^*$-algebra $C^*_rΓ$ and its injective envelope $I(C^*_rΓ)$. In other words, we have natural inclusion $C^*_rΓ\subset C^*_r(Γ,\partialΓ) \subset I(C^*_rΓ)$ of $C^*$-algebras.

math.OA↗
arXiv2007

On a class of $\mathrm{II}_1$ factors with at most one Cartan subalgebra

We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,μ)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy.

math.OA↗
arXiv2005

Boundary Amenability of Relatively Hyperbolic Groups

Let K be a fine hyperbolic graph and G be a group acting on K with finite quotient. We prove that G is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group G acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.

math.GR↗
arXiv2004

Weakly Exact von Neumann Algebras

The theory of exact C*-algebras was introduced by Kirchberg and has been influential in recent development of C*-algebras. A fundamental result on exact C*-algebras is a local characterization of exactness. The notion of weakly exact von Neumann algebras was also introduced by Kirchberg. In this paper, we give a local characterization of weak exactness. As a corollary, we prove that a discrete group is exact if and only if its group von Neumann algebra is weakly exact. The proof naturally involves the operator space duality.

math.OA↗
arXiv2004

About the QWEP conjecture

This is a detailed survey on the QWEP conjecture and Connes' embedding problem. Most of contents are taken from Kirchberg's paper [Invent. Math. 112 (1993)].

math.OA↗
arXiv2004

A note on non-amenability of B(\ell_p) for p=1,2

This is an expository note on non-amenabilty of the Banach algebra B(\ell_p) for p=1,2. These were proved respectively by Connes (p=2) and Read (p=1) via very different methods. We give a single proof which reproves both.

math.FA↗
arXiv2003

Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

Let $\ell$ be a length function on a group G, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that if G is a hyperbolic group and if $\ell$ is a word-length function on G, then the topology from this metric coincides with the weak-* topology (our definition of a ``compact quantum metric space''). We show that a convenient framework is that of filtered $C^*$-algebras which satisfy a suitable `` Haagerup-type'' condition. We also use this framework to prove an analogous fact for certain reduced free products of $C^*$-algebras.

math.OA↗
arXiv2003

Some prime factorization results for type II_1 factors

We prove several unique prime factorization results for tensor products of type II_1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if $R \otimes LF_{r_1} \otimes ... \otimes LF_{r_m}$ is isomorphic to a subfactor in $R \otimes LF_{s_1} \otimes >... \otimes LF_{s_n}$, for some $2\leq r_i, s_j \leq \infty$, then $m\le n$.

math.OA↗
arXiv2003

Solid von Neumann Algebras

We prove that the relative commutant of a diffuse von Neumann subalgebra in a hyperbolic group von Neumann algebra is always injective. It follows that any non-injective subfactor in a hyperbolic group von Neumann algebra is non-Gamma and prime. The proof is based on C*-algebra theory.

math.OA↗
arXiv2002

There is no separable universal II_1-factor

Gromov constructed uncountably many pairwise non-isomorphic discrete groups with Kazhdan's property (T). We will show that no separable II_1-factor can contain all these groups in its unitary group. In particular, no separable II_1-factor can contain all separable II_1-factors in it. We also show that the full group C*-algebras of some of these groups fail the lifting property.

math.OA↗