A properly infinite C*-algebra which is not K_1-injective

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…

真无限因子中的Voiculescu定理

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。

Deformations of infinite projections

Let $A=(A_x)$ be a (semi-)continuous field of $C^*$-algebras over a compact Hausdorff space $X$ and let $p=(p_x)$ be a projection in $A$ such that each $p_x\in A_x$ is properly infinite ($x\in X$). Then $p$ is properly infinite if the field $A$ is upper semi-continuous. However, $p$ can be finite if the field is lower semi-continuous.

Algebraic K-theory and properly infinite C*-algebras

We show that several known results about the algebraic K-theory of tensor products of algebras with the C*-algebra of compact operators in Hilbert space remain valid for tensor products with any properly infinite C*-algebra.

Purely infinite corona algebras

Let A be a simple, sigma-unital, non-unital C*-algebra, with metrizable tracial simplex T(A), which is projection-surjective and injective and has strict comparison of positive elements by traces. Then the following are equivalent: (i) A has quasicontinuous scale; (ii) The multiplier algebra M(A) has strict comparison of positive elements by traces; (iii) The coronal algebra M(A)/A is purely infinite; (iii') The quotient M(A)/Imin is purely infinite; (iv) M(A) has finitely many ideals; (v) Imin=Ifin. If furthermore algebra of n by n matrices of A is projection-surjective and injective for ever…

On von Neumann algebras which are complemented subspaces of B(H)

If there exists a completely bounded projection of B(H) onto a von Neumann algebra M on H, then M is injective. If there exists a bounded projection and M is properly infinite, the same conclusion holds.

真无限C*-代数中的李理想

Lie ideals in properly infinite C*-algebras

该研究证明含单位元的真无限C*-代数中李理想与唯一双边理想交换子等价，得到实秩零无特征标C*-代数的类似结果，并应用于von Neumann代数解决了相关公开问题。

张量范畴对C$^*$-代数的正确外作用

Properly Outer Actions of Tensor Categories on C$^*$-algebras

研究有限指数的外同态和双模对简单C$^*$-代数的正确外性，证明有限指数外同态自动满足正确外性，从而得出单位ary张量范畴对简单C$^*$-代数的自由性。

Purely infinite crossed products by endomorphisms

We study the crossed product $C^*$-algebra associated to injective endomorphisms, which turns out to be equivalent to study the crossed product by the dilated autormorphism. We prove that the dilation of the Bernoulli $p$-shift endomorphism is topologically free. As a consequence, we have a way to twist any endomorphism of a $\D$-absorbing $C^*$-algebra into one whose dilated automorphism is essentially free and have the same $K$-theory map than the original one. This allows us to construct purely infinite crossed products $C^*$-algebras with diverse ideal structures.

The Corona Factorization Property and Refinement Monoids

The Corona Factorization Property of a C*-algebra, originally defined to study extensions of C*-algebras, has turned out to say something important about intrinsic structural properties of the C*-algebra. We show in this paper that a σ-unital C*-algebra A of real rank zero has the Corona Factorization roperty if and only if its monoid V(A) of Murray-von Neumann equivalence classes of projections in matrix algebras over A has a certain (rather weak) comparability property that we call the Corona Factorization Property (for monoids). We show that a projection in such a C*-algebra is properly inf…

一个非 $K_1$-内射的简单 C$^*$-代数

A simple C$^*$-algebra which is not $K_1$-injective

本文构造了一个单的单位元无限维 C$^*$-代数，其 $K_1$-内射性失败，通过推广 Villadsen 的矩阵陷阱技术，利用 $\pi_4(U(2))$ 的生成元作为种子，并借助自旋配边障碍完成证明。

Universal C*-algebras of endomorphisms of groups and partial actions

We present a different way to study the C*-algebra associated with an injective endomorphism of a group G of infinite cokernel. We follow the work of Boava and Exel to construct a partial crossed product representation of that C*-algebra and show properties only using such tools.

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.

Spatiality of derivations on the algebra of $τ$-compact operators

This paper is devoted to derivations on the algebra $S_0(M, τ)$ of all $τ$-compact operators affiliated with a von Neumann algebra $M$ and a faithful normal semi-finite trace $τ.$ The main result asserts that every $t_τ$-continuous derivation $D:S_0(M, τ)\rightarrow S_0(M, τ)$ is spatial and implemented by a $τ$-measurable operator affiliated with $M$, where $t_τ$ denotes the measure topology on $S_0(M, τ)$. We also show the automatic $t_τ$-continuity of all derivations on $S_0(M, τ)$ for properly infinite von Neumann algebras $M$. Thus in the properly infinite case the condition of $t_τ$-cont…

A note on crossed products of rotation algebras

We compute the $K$-theory of crossed products of rotation algebras $\mathcal{A}_θ$, for any real angle $θ$, by matrices in $\mathrm{SL}(2,\mathbb{Z})$ with infinite order. Using techniques of continuous fields, we show that the canonical inclusion of $\mathcal{A}_θ$ into the crossed products is injective at the level of $K_0$-groups. We then give an explicit set of generators for the $K_0$-groups and compute the tracial ranges concretely.

Discretization of C*-algebras

We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra $A$ is a $*$-homomorphism $A \to M$ that factors through the canonical inclusion $C(X) \subseteq \ell^\infty(X)$ when restricted to a commutative C*-subalgebra. Any C*-algebra admits an injective but nonfunctorial discretization, as well as a possibly noninjective functorial discretization, where $M$ is a C*-algebra. Any subhomogenous C*-algebra admits an injective functorial discretization, where $M$ is a W*-algebra. However, any functorial discretization, where $M$ is an AW*-a…

Crossed product decompositions of a purely infinite von Neumann algebra with faithful, almost periodic weight

For $\MvN$ a separable, purely infinite von Neumann algebra with almost periodic weight $ϕ$, a decomposition of $\MvN$ as a crossed product of a semifinite von Neumann algebra by a trace--scaling action of a countable abelian group is given. Then Takasaki's continuous decomposition of the same algebra is related to the above discrete decomposition via Takesaki's notion of induced action, but here one induces up from a dense subgroup. The above results are used to give a model for the one--parameter trace--scaling action of $\Real_+$ on the injective II$_\infty$ factor. Finally, another model o…

C*-algebras associated with endomorphisms of groups

In this work we construct a C*-algebra from an injective endomorphisms of some group G, allowing the endomorphism to have infinite cokernel. We generalize results obtained by I. Hirshberg and also by J. Cuntz and A. Vershik. In good cases we show that the C*-algebra that we study is classifiable by Kirchberg's classification theorem, with K-groups equal to K*(C*(G)).