An infinite family of non-isomorphic C*-algebras with identical K-theory

An Uncountable Family of Non Orbit Equivalent Actions of $\Bbb F_n$

For each $2 \leq n \leq \infty$, we construct an uncountable family of free ergodic measure preserving actions $α_t$ of the free group $\Bbb F_n$ on the standard probability space $(X, μ)$ such that any two are non orbit equivalent (in fact, not even stably orbit equivalent). These actions are all ``rigid'' (in the sense of [Po01]), with the II$_1$ factors $L^\infty(X, μ)\rtimes_{α_t} \Bbb F_n$ mutually non stably isomorphic (even non-stably isomorphic) and in the class $\Cal H\Cal T_{_{s}}.$

Lovász theta type norms and Operator Systems

To each graph on $n$ vertices there is an associated subspace of the $n \times n$ matrices called the operator system of the graph. We prove that two graphs are isomorphic if and only if their corresponding operator systems are unitally completely order isomorphic. This means that the study of graphs is equivalent to the study of these special operator systems up to the natural notion of isomorphism in their category. We define new graph theory parameters via this identification. Certain quotient norms that arise from studying the operator system of a graph give rise to a new family of paramet…

A family of 2-graphs arising from two-dimensional subshifts

Higher-rank graphs (or $k$-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz-Krieger $C^*$-algebras of Robertson and Steger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse the $C^*$-algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute their $K$-theory. We find examples whose $C^*$-algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic …

C*-algebras generated by groups of composition operators

We compute the C*-algebra generated by a group of composition operators acting on certain reproducing kernel Hilbert spaces over the disk, where the symbols belong to a non-elementary Fuchsian group. We show that such a C*-algebra contains the compact operperators, and its quotient is isomorphic to the crossed product C*-algebra determined by the action of the group on the boundary circle. In addition we show that the C*-algebras obtained from composition operators acting on a natural family of Hilbert spaces are in fact isomorphic, and also determine the same Ext-class, which can be related t…

III₁型因子的新构造

A new construction of factors of type ${\rm III_1}$

本文构造了一族新的III₁型因子，并在一定条件下借助Popa的结果给出完全分类，同时构造出具有相同τ不变量但两两不同构的不可数III₁型因子族。

Free products in AQFT

We apply the free product construction to various local algebras in algebraic quantum field theory. If we take the free product of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism, we obtain a half-sided modular inclusion with ergodic canonical endomorphism and trivial relative commutant. On the other hand, if we take Möbius covariant nets with trace class property, we are able to construct an inclusion of free product von Neumann algebras with large relative commutant, by considering either a finite family of identical inclusions or an infinite famil…