More Like this

1. Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields

   https://doi.org/10.1515/crelle-2022-0053  

   Grantcharov, Dimitar; Serganova, Vera (September 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝒲 n {{\mathcal{W}}_{n}} be the Lie algebra of polynomial vector fields.We classify simple weight 𝒲 n {{\mathcal{W}}_{n}} -modules M with finite weight multiplicities. We prove that every such nontrivial module M is either a tensor module or the unique simple submodule in a tensor module associatedwith the de Rham complex on ℂ n {\mathbb{C}^{n}} . 

   more » « less
2. On classification of non-unital amenable simple C*-algebras, II

   https://doi.org/doi.org/10.1016  

   Gong, G.; Lin, Huaxin (December 2020, Journal of geometry and physics)

   null (Ed.)

   We present a classification theorem for separable amenable simple stably projectionless C -algebras with finite nuclear dimension whose K0 vanish on traces which satisfy the Universal Coefficient Theorem. One of C -algebras in the class is denoted by Z0 which has a unique tracial state, K_0(Z_0) = Z and K1(Z_0) = {0}. Let A and B be two separable amenable simple C -algebras satisfying the UCT. We show that A ⊗ Z_0 = B ⊗ Z_0 if and only if Ell(A ⊗ Z_0 ) = Ell(B ⊗ Z_0 ). A class of simple separable C -algebras which are approximately sub-homogeneous whose spectra having bounded dimension is shown to exhaust all possible Elliott invariant for C -algebras of the form A ⊗ Z_0 , where A is any finite separable simple amenable C -algebras. 

   more » « less
3. Unitary groups and augmented Cuntz semigroups of separable simple Z-Stable C∗-algebras

   https://doi.org/10.1142/S0129167X22500185  

   Lin, Huaxin (February 2022, International journal of mathematics)

   Let A be a separable simple exact Z-stable C∗-algebra. We show that the unitary group of \tilde{A} has the cancellation property. If A has continuous scale then the Cuntz semigroup of A has strict comparison property and a weak cancellation property. Let C be a 1-dimensional noncommutative CW complex with K1(C) = {0}. Suppose that λ : Cu∼(C) → Cu∼(A) is a morphism in the augmented Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms φn : C → A such that limn→∞ Cu∼(φn) = λ. This result leads to the proof that every separable amenable simple C∗-algebra in the UCT class has rationally generalized tracial rank at most one. 

   more » « less
4. Covering entropy for types in tracial W^*-algebras

   https://doi.org/10.4115/jla.2023.15.2  

   Jekel, David (June 2023, Journal of Logic and Analysis)

   We study embeddings of tracial $$\mathrm{W}^*$$-algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques.  Jung implicitly and Hayes explicitly defined \emph{$$1$$-bounded entropy} through the asymptotic covering numbers of Voiculescu's microstate spaces, that is, spaces of matrix tuples $$(X_1^{(N)},X_2^{(N)},\dots)$$ having approximately the same $$*$$-moments as the generators $$(X_1,X_2,\dots)$$ of a given tracial $$\mathrm{W}^*$$-algebra.  We study the analogous covering entropy for microstate spaces defined through formulas that use suprema and infima, not only $$*$$-algebra operations and the trace | formulas such as arise in the model theory of tracial $$\mathrm{W}^*$$-algebras initiated by Farah, Hart, and Sherman.  By relating the new theory with the original $$1$$-bounded entropy, we show that if $$\mathcal{M}$$ is a separable tracial $$\mathrm{W}^*$$-algebra with $$h(\cN:\cM) \geq 0$$, then there exists an embedding of $$\cM$$ into a matrix ultraproduct $$\cQ = \prod_{n \to \cU} M_n(\C)$$ such that $$h(\cN:\cQ)$$ is arbitrarily close to $$h(\cN:\cM)$$.  We deduce that if all embeddings of $$\cM$$ into $$\cQ$$ are automorphically equivalent, then $$\cM$$ is strongly $$1$$-bounded and in fact has $$h(\cM) \leq 0$$. 

   more » « less
5. The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension

   https://doi.org/10.1016/j.geomphys.2020.103861  

   Elliott, G.A; Gong, G.; Lin, Huaxin; Niu, Z. (December 2020, Journal of geometry and physics)

   null (Ed.)

   The class of simple separable KK-contractible (KK-equivalent to \{0\} ) C*-algebra s which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras A\otimes K is classifiable, where A is a simple separable C*-algebra with finite nuclear dimension and is the simple inductive limit of Razak algebras with unique trace, which is bounded 

   more » « less