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Narutaka Ozawa

Publications and source records attributed to Narutaka Ozawa.

At least 37 records · Page 2Linked to original sources
arXiv2016

C*-simplicity and the unique trace property for discrete groups

A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take the study of C*-simplicity a step further, and in addition to settle the longstanding open problem of characterizing groups with the unique trace property. We give a new and self-contained proof of the aforementioned characterization of C*-simplicity. This yields a new characterization of C*-simplicity in terms of the weak containment of quasi-regular representations. We introduce a convenient algebraic condition that implies C*-simplicity, and show that this condition is satisfied by a vast class of groups, encompassing virtually all previously known examples as well as many new ones. We also settle a question of Skandalis and de la Harpe on the simplicity of reduced crossed products. Finally, we introduce a new property for discrete groups that is closely related to C*-simplicity, and use it to prove a broad generalization of a theorem of Zimmer, originally conjectured by Connes and Sullivan, about amenable actions.

math.OA↗
arXiv2015

Real positivity and approximate identities in Banach algebras

Blecher and Read have recently introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the present paper we generalize some part of this, and some other facts, to larger classes of Banach algebras.

math.FA↗
arXiv2015

Haagerup approximation property via bimodules

The Haagerup approximation property (HAP) is defined for finite von Neumann algebras in such a way that the group von Neumann algebra of a discrete group has the HAP if and only if the group itself has the Haagerup property. The HAP has been studied extensively for finite von Neumann algebras and it is recently generalized for arbitrary von Neumann algebras by Caspers-Skalski and Okayasu-Tomatsu. One of the motivations behind the generalization is the fact that quantum group von Neumann algebras are often infinite even though the Haagerup property has been defined successfully for locally compact quantum groups by Daws-Fima-Skalski-White. In this paper, we partly fill this gap by proving that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the HAP. This is new even for genuine locally compact groups.

math.OA↗
arXiv2015

Dixmier approximation and symmetric amenability for C*-algebras

We study some general properties of tracial C*-algebras. In the first part, we consider Dixmier type approximation theorem and characterize symmetric amenability for C*-algebras. In the second part, we consider continuous bundles of tracial von Neumann algebras and classify some of them.

math.OA↗
arXiv2015

Noncommutative real algebraic geometry of Kazhdan's property (T)

It is well-known that a finitely generated group $Γ$ has Kazhdan's property (T) if and only if the Laplacian element $Δ$ in ${\mathbb R}[Γ]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in ${\mathbb R}[Γ]$. Namely, $Γ$ has property (T) if and only if there are a constant $κ>0$ and a finite sequence $ξ_1,...,ξ_n$ in ${\mathbb R}[Γ]$ such that $Δ^2-κΔ= \sum_i ξ_i^*ξ_i$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in ${\mathbb R}[Γ]$, possibly with an assist of computers.

math.GR↗
arXiv2014

Elementary amenable groups are quasidiagonal

We show that the group C*-algebra of any elementary amenable group is quasidiagonal. This is an offspring of recent progress in the classification theory of nuclear C*-algebras.

math.OA↗
arXiv2014

A continuum of $\mathrm{C}^*$-norms on $\IB(H)\otimes \IB(H)$ and related tensor products

For any pair $M,N$ of von Neumann algebras such that the algebraic tensor product $M\otimes N$ admits more than one $\mathrm{C}^*$-norm, the cardinal of the set of $\mathrm{C}^*$-norms is at least $ {2^{\aleph_0}}$. Moreover there is a family with cardinality $ {2^{\aleph_0}}$ of injective tensor product functors for $\mathrm{C}^*$-algebras in Kirchberg's sense. Let $\IB=\prod_n M_{n}$. We also show that, for any non-nuclear von Neumann algebra $M\subset \IB(\ell_2)$, the set of $\mathrm{C}^*$-norms on $\IB \otimes M$ has cardinality equal to $2^{2^{\aleph_0}}$.

math.OA↗
arXiv2014

Group approximation in Cayley topology and coarse geometry, Part III: Geometric property (T)

In this series of papers, we study correspondence between the following: (1) large scale structure of the metric space bigsqcup_m {Cay(G(m))} consisting of Cayley graphs of finite groups with k generators; (2) structure of groups which appear in the boundary of the set {G(m)}_m in the space of k-marked groups. In this third part of the series, we show the correspondence among the metric properties `geometric property (T),' `cohomological property (T),' and the group property `Kazhdan's property (T).' Geometric property (T) of Willett--Yu is stronger than being expander graphs. Cohomological property (T) is stronger than geometric property (T) for general coarse spaces.

math.OA↗
arXiv2013

A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra

It has been a longstanding problem whether every amenable operator algebra is isomorphic to a (necessarily nuclear) C*-algebra. In this note, we give a nonseparable counterexample. The existence of a separable counterexample remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in C*-algebras and show that our method cannot produce a separable counterexample.

math.OA↗
arXiv2013

Tsirelson's Problem and Asymptotically Commuting Unitary Matrices

In this note, we consider quantum correlations of bipartite systems having a slight interaction, and reinterpret Tsirelson's problem (and hence Kirchberg's and Connes's conjectures) in terms of finite-dimensional asymptotically commuting positive operator valued measures. We also consider the systems of asymptotically commuting unitary matrices and formulate the Stronger Kirchberg Conjecture.

math.OA↗
arXiv2013

About the Connes Embedding Conjecture---Algebraic approaches---

This is an expanded lecture note for "Masterclass on sofic groups and applications to operator algebras" (University of Copenhagen, 5-9 November 2012). It is about algebraic aspects of the Connes Embedding Conjecture. It contains new proofs of equivalence of the Connes Embedding Conjecture, Positivstellensatze for trace positive polynomials, Kirchberg's Conjecture, and Tsirelson's Problem.

math.OA↗
arXiv2012

Free Banach spaces and the approximation properties

We characterize the metric spaces whose free space has the bounded approximation property through a Lipschitz analogue of the local reflexivity principle. We show that there exist compact metric spaces whose free spaces fail the approximation property.

math.FA↗
arXiv2012

Is an irng singly generated as an ideal?

Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (abbreviated as irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for perfect groups is discussed.

math.RA↗
arXiv2011

A remark on contractible Banach algebras

It is a longstanding problem whether every contractible Banach algebra is necessarily finite-dimensional. In this note, we confirm this for Banach algebras acting on Banach spaces with the uniform approximation property. This generalizes a result of Paulsen and Smith who proved the same for Banach algebras acting on Hilbert spaces.

math.FA↗
arXiv2011

Examples of groups which are not weakly amenable

We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non weak amenability results of Haagerup (1988) and Ozawa--Popa (2010). A von Neumann algebra analogue is also obtained.

math.OA↗