Injective Envelopes of Separable C*-algebras

Real structure in operator spaces, injective envelopes and $G$-spaces

We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and $C^*$-envelope. We consider the interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group.

$C^*$-simplicity and Ozawa conjecture for groupoid $C^*$-algebras, part I: injective envelopes

This paper studies injective envelopes of groupoid dynamical systems and the corresponding boundaries. Analogue to the group case, we associate a bundle of compact Hausdorff spaces to any (discrete) groupoid (the Hamana boundary of the groupoid). We study bundles of compact topological spaces equipped with an action of a groupoid. We show that any groupoid has a minimal boundary (the Furstenberg boundary of the groupoid). We prove that the Hamana and Furstenberg boundaries are the same, for (discrete) groupoids. We find the relation between the reduced crossed product of the $\G$-injective env…

实 C*-代数与 AW*-代数的内射包络

Injective envelopes of real C*- and AW*-algebras

本文研究实 C*-代数的内射包络，证明外 *-自同构可唯一延拓、单实 C*-代数的内射包络为单实 AW*-因子，并构造例子说明 III 型内射实 AW*-因子类比对应 W*-因子类至少多一个。

Purity of the Embeddings of Operator Systems into their C$^*$- and Injective Envelopes

We study the issue of issue of purity (as a completely positive linear map) for identity maps on operators systems and for their completely isometric embeddings into their C$^*$-envelopes and injective envelopes. Our most general result states that the canonical embedding of an operator system $\mathcal R$ into its injective envelope $\mathcal I(\mathcal R)$ is pure if and only if the C$^*$-envelope of $\mathcal R$ is a prime C$^*$-algebra. To prove this, we also show that the identity map on any AW$^*$-factor is a pure completely positive linear map. For embeddings of operator systems into th…

On the uniqueness of injective III$_1$ factor

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.

算子空间内射包存在的简短证明

A short proof of the existence of the injective envelope of an operator space

本文利用Ellis引理，为算子空间内射包的存在性提供了一种简单的证明，简化了Hamana和Ruan原有的工作。

关于内射包络与冯·诺伊曼代数的一个注记

A note on injective envelopes and von Neumann algebras

本文探讨了 C*-代数 A 的内射包络 I(A) 与其内射表示 lambda 生成的冯·诺伊曼代数之间的关系，证明了存在接近 *-同态的正收缩映射，其核为双边理想。

局部算子空间的R-希洛夫边界

The R-Shilov boundary for a local operator space

本文研究了局部算子空间的R-希洛夫边界，定义了注入R-包和R-C*-包，并证明了其与Dosi（2014）的构造一致。

Maximal C*-algebras of quotients and injective envelopes of C*-algebras

A new C*-enlargement of a C*-algebra $A$ nested between the local multiplier algebra $M_{\text{loc}}(A)$ of $A$ and its injective envelope $I(A)$ is introduced. Various aspects of this maximal C*-algebra of quotients, $Q_{\text{max}}(A)$, are studied, notably in the setting of AW*-algebras. As a by-product we obtain a new example of a type I C*-algebra $A$ such that $M_{\text{loc}}(M_{\text{loc}}(A))\ne M_{\text{loc}}(A)$.

部分 C* - 动力系统的内射包

Injective envelopes of partial C*-dynamical systems

该研究将滨名内射包理论扩展到部分 C* - 动力系统领域，证明了系统与内射包理想交性质的关系，建立了系统内射包与其包络作用内射包间的联系，还表明对阿贝尔系统构造与相关函数代数一致，关键是考虑了广义系统概念。

Jordan格完备化及关于内射包络与von Neumann代数的注记

The Jordan lattice completion and a note on injective envelopes and von Neumann algebras

本文通过为函数系引入两种格构造，研究可分C*-代数的内射包络与包络von Neumann代数的关系，并刻画了由正投影上的取值决定投影映射的非线性映射。

Multipliers of operator spaces, and the injective envelope

We study the injective envelope I(X) of an operator space X, showing amongst other things that it is a self-dual C$^*-$module. We describe the diagonal corners of the injective envelope of the canonical operator system associated with X. We prove that if X is an operator $A-B$-bimodule, then A and B can be represented completely contractively as subalgebras of these corners. Thus, the operator algebras that can act on X are determined by these corners of I(X) and consequently bimodules actions on X extend naturally to actions on I(X). These results give another characterization of the multipli…

作为算子模的$C^*$-代数的内射包络

Injective Envelopes of $C^*$-algebras as Operator Modules

本文在算子空间框架下刻画了$C^*$-代数的内射包络$I(A)$，证明其对完全有界$A$-模映射具有刚性，并由此将局部乘子代数自然嵌入$I(A)$中，简化并推广了相关已知结果。

On maximal injective subalgebras of tensor products of von Neumann algebras

Let M_i be a von Neumann algebra, and B_i be a maximal injective von Neumann subalgebra of M_i, i=1,2. If M_1 has separable predual and the center of B_1 is atomic, e.g., B_1 is a factor, then B_1\tensor B_2 is a maximal injective von Neuamnn subalgebra of M_1\tensor M_2. This partly answers a question of Popa

C*-代数的内射包络与局部乘子代数

Injective envelopes and local multiplier algebras of C*-algebras

本文研究C*-代数的局部乘子代数与内射包络的关系，证明前者可嵌入后者、二者内射包络一致，并给出Wittstock型延拓定理成立当且仅当二者相等，且该问题等价于Kaplansky关于AW*-代数单调完备性的猜想。

Hamana的注射包作为最大刚性乘法覆盖

Hamana's injective envelope as a maximal rigid multiplier cover

本文研究了在Hamana意义下刚性的乘法覆盖，证明Hamana注射包是最大刚性乘法覆盖，并揭示了其在序理论中的 characterization。

Maximal injective and mixing masas in group factors

We present families of pairs of finite von Neumann algebras $A\subset M$ where $A$ is a maximal injective masa in the type $\mathrm{II}_1$ factor $M$ with separable predual. Our results make use of the strong mixing and the asymptotic orthogonality properties of $A$ in $M$ and are borrowed from ideas of S. Popa who proved that if $G$ is a non abelian free group and if $a$ is one of its generators, then the von Neumann algebra generated by $a$ is maximal injective in the factor $L(G)$. Our results apply to pairs $H<G$ where $H$ is an infinite abelian subgroup of a suitable amalgamated product g…

内射型III$_{1}$因子上极端几乎周期态的唯一性

Uniqueness of extremal almost periodic states on the injective type III$_{1}$ factor

本文证明了内射型III$_{1}$因子上具有相同点谱的极端几乎周期态在自同构作用下唯一，并据此将其参数化，同时应用于Cuntz代数KMS态的分类。

Semicrossed products of C*-algebras and their C*-envelopes

Let $\mathcal{C}$ be a C*-algebra and $α:\mathcal{C} \rightarrow \mathcal{C}$ a unital *-endomorphism. There is a natural way to construct operator algebras which are called semicrossed products, using a convolution induced by the action of $α$ on $\mathcal{C}$. We show that the C*-envelope of a semicrossed product is (a full corner of) a crossed product. As a consequence, we get that, when $α$ is *-injective, the semicrossed products are completely isometrically isomorphic and share the same C*-envelope, the crossed product $\mathcal{C}_\infty \rtimes_{α_\infty} \mathbb{Z}$. We show that mini…

k1-injectivity of the Paschke dual algebra for certain simple C*-algebras

Let $\mathcal{B}$ be a nonunital separable simple stable C*-algebra with strict comparison of positive elements and $T(\mathcal{B})$ having finite extreme boundary, and let $\mathcal{A}$ be a simple unital separable nuclear C*-algebra. We prove that the Paschke dual algebra $\mathcal{A}^d_{\mathcal{B}}$ is $K_1$-injective. As a consequence, we obtain interesting $KK$-uniqueness theorems which generalize the Brown-Douglas-Fillmore essential codimension property.

Semicrossed Products of Operator Algebras by Semigroups

We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive endomorphisms. The choice of allowable representations affects the corresponding universal algebra. We seek quite general conditions which will allow us to show that the C*-envelope of the semicrossed product is (a full corner of) a crossed product of an auxiliary C*-algebra by a group action. Our analysis concerns a case-by-case dilation theory on covariant pairs. In the process we determine the C*-envelope fo…

Trivialization of C(X)-algebras with strongly self-absorbing fibres

Suppose $A$ is a separable unital $C(X)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$-injective $C^{*}$-algebra $D$. We show that $A$ and $C(X) \otimes D$ are isomorphic as $C(X)$-algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

不存在通用可分巴拿赫代数

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。

Quasimultipliers of Operator Spaces

We use the injective envelope to study quasimultipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasimultipliers. We obtain generalizations of the Banach-Stone theorem.

Tensor algebras of product systems and their C*-envelopes

Let $(G, P)$ be an abelian, lattice ordered group and let $X$ be a compactly aligned product system over $P$. We show that the C*-envelope of the Nica tensor algebra $\mathcal{N}\mathcal{T}^+_X$ coincides with both Sehnem's covariance algebra $\mathcal{A} \times_X P$ and the co-universal C*-algebra $\mathcal{N}\mathcal{O}^r_X$ for injective, gauge compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of $\mathcal{N}\math…

Real Operator Algebras and Real Completely Isometric Theory

This paper is a continuation of the program started by Ruan in 2003, of developing real operator space theory. In particular, we develop the theory of real operator algebras. We also show among other things that the injective envelope, C*-envelope and non-commutative Shilov boundary exist for a real operator space. We develop real one-sided M-ideal theory and characterize one-sided M-ideals in real C*-algebras and real operator algebras with contractive approximate identity.

Classification of hyperfinite factors up to completely bounded isomorphism of their preduals

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…

Contributions to the theory of C*-correspondences with applications to multivariable dynamics

Motivated by the theory of tensor algebras and multivariable C*-dynamics, we revisit two fundamental techniques in the theory of C*-correspondences, the "addition of a tail" to a non-injective C*-correspondence and the dilation of an injective C*-correspondence to an essential Hilbert bimodule. We provide a very broad scheme for "adding a tail" to a non-injective C*-correspondence; our scheme includes the "tail" of Muhly and Tomforde as a special case. We illustrate the diversity and necessity of our tails with several examples from the theory of multivariable C*-dynamics. We also exhibit a tr…

看似内射的冯·诺依曼代数

Seemingly injective von Neumann algebras

本文证明 QWEP 冯·诺依曼代数具有弱*正逼近性质当且仅当它是看似内射的，并推导出具有可分预对偶的代数同构于 $B(\ell_2)$，同时指出因 $B(H)$ 缺乏逼近性质而存在非看似内射的反例。

Semicrossed products of operator algebras and their C*-envelopes

Let $\A$ be a unital operator algebra and let $α$ be an automorphism of $\A$ that extends to a *-automorphism of its $\ca$-envelope $\cenv (\A)$. In this paper we introduce the isometric semicrossed product $\A \times_α^{\is} \bbZ^+ $ and we show that $\cenv(\A \times_α^{\is} \bbZ^+) \simeq \cenv (\A) \times_α \bbZ$. In contrast, the $\ca$-envelope of the familiar contractive semicrossed product $\A \times_α \bbZ^+ $ may not equal $\cenv (\A) \times_α \bbZ$. Our main tool for calculating $\ca$-envelopes for semicrossed products is the concept of a relative semicrossed product of an operator al…

C*-envelopes of tensor algebras of product systems

Let $P$ be a submonoid of a group $G$ and let $\mathcal{E}=(\mathcal{E}_p)_{p\in P}$ be a product system over $P$ with coefficient C*-algebra $A$. We show that the following C*-algebras are canonically isomorphic: the C*-envelope of the tensor algebra $\mathcal{T}_λ(\mathcal{E})^+$ of $\mathcal{E}$; the reduced cross sectional C*-algebra of the Fell bundle associated to the canonical coaction of $G$ on the covariance algebra $A\times_{\mathcal{E}}P$ of $\mathcal{E}$; and the C*-envelope of the cosystem obtained by restricting the canonical gauge coaction on $\mathcal{T}_λ(\mathcal{E})$ to the …

A weak expectation property for operator modules, injectivity and amenable actions

We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras $A$. We prove a number of general results---for example, a characterization of the $A$-WEP in terms of an appropriate $A$-injective envelope, and also a characterization of those $A$ for which $A$-WEP implies WEP. In the case of $A=L^1(G)$, we recover the $G$-WEP for $G$-$C^*$-algebras in recent work of Buss--Echterhoff--Willett. When $A=A(G)$, we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related…

Rokhlin actions and self-absorbing C*-algebras

Let A be a unital separable C*-algebra, and D a K_1-injective strongly self-absorbing C*-algebra. We show that if A is D-absorbing, then the crossed product of A by a compact second countable group or by Z or by R is D-absorbing as well, assuming the action satisfying a Rokhlin property. In the case of a compact Rokhlin action we prove a similar statement about approximate divisibility.

具有量子对称性的代数与范畴的内射性

Injectivity for algebras and categories with quantum symmetry

本文证明了含幺Yetter-Drinfeld C*-代数及相关双模范畴的内射包存在性，并由此得出紧量子群Drinfeld double边界作用的幺单不变性。

Embedding of exact C*-algebras and continuous fields in the Cuntz algebra O_2

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra ${\cal O}_2.$ We further prove that if $A$ is a simple separable unital nuclear C*-algebra, then ${\cal O}_2 \otimes A \cong {\cal O}_2,$ and if, in addition, $A$ is purely infinite, then ${\cal O}_{\infty} \otimes A \cong A.$ The embedding of exact C*-algebras in $\OA{2}$ is continuous in the following sense. If $A$ is a continuous field of C*-algebras over a compact manifold or finite CW complex $X$ with fiber $A (x)$ over $x \in X,$ such that the algebra of continuous sections of $A$ is separable…