Narutaka Ozawa
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.
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It is well-known that every commutative separable unital C*-algebra of real rank zero is a quotient of the C*-algebra of all compex continous functions defined on the Cantor cube. We prove a non-commutative version of this result by showing that the class of all separable unital C*-algebras of real rank zero concides with the class of quotients of a certain separable unital C*-algebra of real-rank zero.
There is no universal separable Banach algebra
该研究证明不存在对所有可分巴拿赫代数同态嵌入通用的可分巴拿赫代数，在交换范畴也成立，通过构造$A_X$并结合Johnson和Szankowski的定理完成简短直接的证明。
We show a classification theorem for locally compact separable abelain group actions on factorsw with the Rohlin property. This is a generalization of Masuda and Tomatsu's work.
We prove that the infinite tensor power of a unital separable C*-algebra absorbs the Jiang-Su algebra Z tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C*-algebras without characters known to lie outside it. This category moreover contains the vast majority of our stock-in-trade separable amenable C*-algebras, and is closed under passage to separable superalgebras and quotients, and hence to unital tensor products, unital direct limits, and cr…
The separable case of Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors
本文基于非交换Lyapunov定理，证明了可分的有限von Neumann代数具有正交基，从而确认了Kadison问题的分离情况。
In 1960 Pukánszky introduced an invariant associating to every masa in a separable $\mathrm{II}_1$ factor a non-empty subset of $\mathbb N\cup\{\infty\}$. This invariant examines the multiplicity structure of the von Neumann algebra generated by the left-right action of the masa. In this paper it is shown that every non-empty subset of $\mathbb N\cup\{\infty\}$ arises as the Pukánszky invariant of some masa in a separable McDuff $\mathrm{II}_1$ factor which contains a masa with Pukánszky invariant $\{1\}$. In particular the hyperfinite $\mathrm{II}_1$ factor and all separable McDuff $\mathrm{I…
In this paper we investigate operator Hilbert systems and their separable morphisms. We prove that the operator Hilbert space of Pisier is an operator system, which possesses the self-duality property. It is established a link between unital positive maps and Pietch factorizations, which allows us to describe all separable morphisms from an abelian C*-algebra to an operator Hilbert system. Finally, we prove a key property of entanglement breaking maps that involves operator Hilbert systems.
Let $G$ be a metrizable compact group, $A$ a separable C*-algebra and $α$ a strongly continuous action of $G$ on $A$. Provided that $α$ satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in E-theory passes from $A$ to the crossed product C*-algebra $A\rtimes_αG$ and the fixed point algebra $A^α$. This extends a result by Gardella in the case that $G$ is the circle and $A$ is nuclear. For circle actions on separable, unital C*-algebras with the continuous Rokhlin property, we establish a connection between the $E$-theory equivalence class of the coefficie…
We define united KK-theory for real C*-algebras A and B such that A is separable and B is sigma-unital, extending united K-theory in the sense that KK\crt(\R, B) = K\crt(B). United KK-theory contains real, complex, and self-conjugate KK-theory; but unlike unaugmented real KK-theory, it admits a universal coefficient theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK\crt(A,B) can be written as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classi…
In this paper, we prove a non-commutative version of the Weyl-von Neumann theorem for representations of unital, separable AH algebras into countably decomposable, semifinite, properly infinite, von Neumann factors, where an AH algebra means an approximately homogeneous ${\rm C}^{\ast}$-algebra. We also prove a result for approximate summands of representations of unital, separable AH algebras into finite von Neumann factors.
We will show that separable unital AF-algebras whose Bratteli diagrams do not allow converging two nodes into one node, can be classified up to the tensor product with the universal UHF-algebra $\Q$ only by their trace spaces. That is, if $\A$ and $\B$ are such AF-algebras, then $T(A)=T(B)$ if and only if $\A\otimes \Q \cong \B\otimes \Q$.
The K-Theory of a Simple Separable Exact C*-Algebra Not Isomorphic to Its Opposite Algebra
本文构造了不可数个不与其反代数同构的单可分稳定有限幺正精确C*-代数，证明了其K0-群、K1-群和迹态空间有不可数种可能，并证明它们满足Universal Coefficient Theorem。
We give a new proof of a theorem due to Alain Connes, that an injective factor $N$ of type III$_1$ with separable predual and with trivial bicentralizer is isomorphic to the Araki--Woods type III$_1$ factor $R_{\infty}$. This, combined with the author's solution to the bicentralizer problem for injective III$_1$ factors provides a new proof of the theorem that up to $*$-isomorphism, there exists a unique injective factor of type III$_1$ on a separable Hilbert space.
We study the approximately finite-dimensional (AF) $C^*$-algebras that appear as inductive limits of sequences of finite-dimensional $C^*$-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra $\mathcal A_\mathfrak{F}$ with the property that any separable AF-algebra is isomorphic to a quotient of $\mathcal A_\mathfrak{F}$. Equivalently, by Elliott's classification of separable AF-algebras, there are surjectively universal countable scaled (or with order-unit) dimension groups. This universality is a consequence of our result stating that $\mathcal A_\mathfr…
We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g.\ $\mathrm{SL}(n,\mathbb{Z}), \: n \geq 3$) and noncommutative Bernoulli shifts with arbitr…
Voiculescu's Theorem in Properly Infinite Factors
本文研究可分真无限因子中的Voiculescu近似酉等价定理，给出可约算子范数稠密性、广义双交换子定理等应用，将结果推广到可分$\text{III}$型因子乘子代数，并引入核长度概念。
Using the classical universal coefficient theorem of Rosenberg-Schochet, we prove a simple classification of all localizing subcategories of the Bootstrap category of separable complex C*-algebras. Namely, they are in bijective correspondence with subsets of the Zariski spectrum of the integers -- precisely as for the localizing subcategories of the derived category of complexes of abelian groups. We provide corollaries of this fact and put it in context with similar classifications available in the literature.
The unitary group of a $\mathrm{II}_1$ factor is SOT-contractible
本研究证明SOT可分$\u2162_1$型因子的酉群在强算子拓扑下可缩，结合旧结果推广至不含有限$\u2160$型直和项的SOT可分冯·诺依曼代数，证明用到自由卷积正则化和Popa定理。
Reducible operators in non-$Γ$ type ${\rm II}_1$ factors
本文研究非$\u0393$型${\ m II}_1$因子中可约算子的范数逼近问题，证明可约算子集在算子范数拓扑下疏朗，提出$\u0393$性质新刻画与单算子谱隙性质作为核心工具。
In this paper we show that the $\mathrm{K}$-homology groups of a separable C*-algebra can be enriched with additional descriptive set-theoretic information, and regarded as definable groups. Using a definable version of the Universal Coefficient Theorem, we prove that the corresponding definable $\mathrm{K}$-homology is a finer invariant than the purely algebraic one, even when restricted to the class of UHF C*-algebras, or to the class of unital commutative C*-algebras whose spectrum is a $1$-dimensional connected subspace of $\mathbb{R}^{3}$.
We prove that any separable II$_1$ factor $M$ admits a {\it coarse decomposition} over the hyperfinite II$_1$ factor $R$, i.e., there exists an embedding $R\hookrightarrow M$ such that $L^2M\ominus L^2R$ is a multiple of the coarse Hilbert $R$-bimodule $L^2R \overline{\otimes} L^2R^{op}$ (equivalently, the von Neumann algebra generated by left and right multiplication by $R$ on $L^2M\ominus L^2R$ is isomorphic to $R\overline{\otimes}R^{op}$). Moreover, if $Q\subset M$ is an infinite index irreducible subfactor, then $R\hookrightarrow M$ can be constructed so that to also be coarse with respect…
In this paper we consider the following problem: When are the preduals of two hyperfinite (=injective) factors $\M$ and $\N$ (on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if $\M$ is semifinite and $\N$ is type III, then their preduals are not cb-isomorphic. Moreover, we construct a one-parameter family of hyperfinite type III$_0$-factors with mutually non cb-isomorphic preduals, and we give a characterization of those hyperfinite factors $\M$ whose preduals are cb-isomorphic to the predual of the unique hyperfinite type III$_1$-factor. In contr…
The similarity of irreducible operators in factors
本文研究了可分因子中与最大阿贝尔自共轭子代数相关的生成元，探讨了可分因子中与不可约算子相似的正规算子的特征。
We find a description of the restriction of doubly stochastic maps to separable abelian $C^*$-subalgebras of a II$_1$ factor $\cM$. We use this local form of doubly stochastic maps to develop a notion of joint majorization between $n$-tuples of mutually commuting self-adjoint operators that extends those of Kamei (for single self-adjoint operators) and Hiai (for single normal operators) in the II$_1$ factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian $C^*$-subalgebra of $\cM$ can be embedded into a separable abelia…
Applying Popa's orthogonality method to a new class of groups, we construct amenable group factors which are prime and have no infinite dimensional regular abelian *-subalgebras. By adjusting Farah--Katsura's solution of Dixmier's problem to the von Neumann algebra setting, we obtain the first examples of prime AFD factors and tensorially prime simple AF-algebras. Our results are proved in ZFC, thus in particular answering questions asked by Farah--Hathaway--Katsura--Tikuisis. We also directly determine central sequences of certain crossed products. This concludes the failure of the Kirchberg …
Classifying $^*$-homomorphisms I: Unital simple nuclear $C^*$-algebras
本文对满足万有系数定理的单位可分核C*-代数到张量吸收Jiang-Su代数的单位单可分核C*-代数的单位嵌入分类，为单位分类定理的稳定有限情形提供新自包含证明。
Products of irreducible operators in factors
本文证明了可分因子中每个算子（除特定类型因子中的零算子外）均可表示为两个不可约算子的乘积，扩展了Radjavi关于算子可表示为两个不可约算子之和的结果。
We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.
We show that stabilizations of sufficiently noncommutative separable unital C*-algebras with finite nuclear dimension have the corona factorization property.
Non-isomorphism of $A^{*n}, 2\leq n \leq \infty$, for a non-separable abelian von Neumann algebra $A$
本文证明非可分阿贝尔迹冯·诺依曼代数的自由幂在 $2\leq n\leq\infty$ 时相互不同构，且有限情形基本群平凡，从而解决自由群因子问题的非可分版本。
We prove that every separable tracial von Neumann algebra embeds into a II$_1$ factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. We also establish an analogous result for the trivial extension over a non-atomic probability space of every countable p.m.p. equivalence relation. These results are obtained by using the class of wreath-like product groups introduced recently in \cite{CIOS21}.
On conjugacy and perturbation of subalgebras
本文研究迹冯·诺依曼代数中子代数的共轭轨道，构造了使具有Haagerup性质或性质(T)的子代数分别具有唯一嵌入的II$_1$因子，并讨论了可分性与密度特征。
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