Simple nuclear C∗-algebras of tracial topological rank one
By Huaxin Lin. 2004
111 citations recorded by the source corpus.
We give a classification theorem for unital separable nuclear C∗-algebras with tracial rank no more than one. Let A and B be two unital separable simple nuclear C∗-algebras with TR(A),TR(B)⩽1 which satisfy the universal coefficient theorem. We show that A≅B if and only if there is an order and unit preserving isomorphism γ=(γ0,γ1,γ2):(K0(A),K0(A)+,[1A],K1(A),T(A))≅(K0(B),K0(B)+,[1B],K1(B),T(B)), where γ2−1(τ)(x)=τ(γ0(x)) for each x∈K0(A) and τ∈T(B).
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5b5e77f04dab9a38b36eb5998f44f1982d8cf911
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Citations in the corpus
- Regularity properties in the classification program for separable amenable C*-algebras (2007, background): “(There is a classification theorem for algebras of tracial rank one ([41]), but this has been somewhat less useful—it is difficult to verify tracial rank one in natural examples.”
- Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras (2008, unknown): “Let 1/2 > ǫ′n > 0 so that Bott(h ′, u′)|Pn is well defined for any unital monomorphism h′ : C → A and any unitary u′ ∈ A for which ‖[h′(a), u′]‖ ǫ′n for all a ∈ Gn (e 7.105) for some finite subset Gn ⊂ ϕn,∞(Cn) which contains Fn.”
- Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras (2014, methodology): “6 of [57]), there is a homomorphism λ : K1(D) → U0(C)/CU(C) such that λ|κ1◦Π‡◦π1(F ) = π0 ◦ ((κ1 ◦ Π ‡ ◦ π1)|F )−1.”
- Crossed products by minimal homeomorphisms (2004, background): “ng order isomorphism Γ(η) 0: K 0(A 1) → K 0(A 2). The state space of K 0(Aj) is exactly the space of quasitraces on Aj (Theorem 6.9.1 of [2]), and it is easy to see from the definition (see 4.9(ix) of [23]) that if a C*-algebra A has tracial rank zero, then all quasitraces on A are in fact traces. So”
- A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. (2018, methodology): “8 of [62]) Let A be a unital C∗-algebra and let B1 be a unital separable simple C∗-algebra in B0 and let B = B1 ⊗ U for some UHF-algebra of infinite type and C ∈ C0 be a C∗-subalgebra of B.”
- The Elliott conjecture for Villadsen algebras of the first type (2006, unknown): “Then, there is a ∗-isomorphism Φ : A → B which induces φ.”
- Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary (2012, result): “ finite-dimensional extreme boundary of trace space, ( in the proof of [4, Lemma 4.4]). This result was essentially based on the works by D. A. Edwards [6], J. Cuntz and G. K. Pedersen [3], and H. Lin [17]. The starting point of our proof of Lemma 4.2 is this proposition. Proposition 4.1. Let A be a ”