arXiv subjects

Liangqing Li

Publications and source records attributed to Liangqing Li.

6 recordsLinked to original sources

arXiv2019

A classification of inductive limit $C^{*}$-algebras with ideal property

Let $A$ be an $AH$ algebra $A=\lim\limits_{n\to \infty}(A_{n}=\bigoplus\limits_{i=1}\limits^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m})$, where $X_{n,i}$ are compact metric spaces, $t_{n}$ and $[n,i]$ are positive integers, and $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. In this article, we will classify all $AH$ algebras with ideal property of no dimension growth---that is, $sup_{n,i}dim(X_{n,i})<+\infty$. This result generalizes and unifies the classification of $AH$ algebras of real rank zero in [EG] and [DG] and the classification of simple $AH$ algebras in [G5] and [EGL1]. This completes one of two important possible generalizations of [EGL1] suggested in the introduction of [EGL1]. The invariants for the classification include the scaled ordered total $K$-group $(\underline{K}(A), \underline{K}(A)_{+},ΣA)$ (as already used in real rank zero case in [DG]), for each $[p]\inΣA$, the tracial state space $T(pAp)$ of cut down algebra $pAp$ with a certain compatibility, (which is used by [Stev] and [Ji-Jiang] for $AI$ algebras with the ideal property), and a new ingredient, the invariant $U(pAp)/\overline{DU(pAp)}$ with a certain compatibility condition, where $\overline{DU(pAp)}$ is the closure of commutator subgroup $DU(pAp)$ of the unitary group $U(pAp)$ of the cut down algebra $pAp$. In [GJL] a counterexample is presented to show that this new ingredient must be included in the invariant. The discovery of this new invariant is analogous to that of the order structure on the total K-theory when one advances from the classification of simple real rank zero $C^*$-algebras to that of non simple real rank zero $C^*$-algebras in [G2], [Ei], [DL] and [DG] (see Introduction below).

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arXiv2019

On the Decomposition Theorems for C*-algebras

Elliott dimension drop interval algebra is an important class among all $C^*$-algebras in the classification theory. Especially, they are building stones of $\mathcal{AHD}$ algebra and the latter contains all $AH$ algebras with the ideal property of no dimension growth. In this paper, we will show two decomposition theorems related to the Elliott dimension drop interval algebra. Our results are key steps in classifying all $AH$ algebras with the ideal property of no dimension growth.

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arXiv2019

On the inductive limit of direct sums of simple TAI algebras

An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive limit $\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus\limits_{i=1}A_{n}^{i},ϕ_{nm})$, where each $A_{n}^{i}$ is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [Jiang1], the second author classified all ATAI algebras by an invariant consisting orderd total K-theory and tracial state spaces of cut down algebras under an extra restriction that all element in $K_{1}(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic $K_{1}$-group as an addition to the invariant used in [Jiang1]. The theorem is proved by reducing the class to the classification theorem of $\mathcal{AHD}$ algebras with ideal property which is done in [GJL1]. Our theorem generalizes the main theorem of [Fa] and [Jiang1] (see corollary 4.3).

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arXiv2019

Hausdorffifized algebraic $K_1$ group and invariants for $C^*$-algebras with the ideal property

A $C^*$-algebra $A$ is said to have the ideal property if each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. $C^*$-algebras with the ideal property are generalization and unification of real rank zero $C^*$-algebras and unital simple $C^*$-algebras. It is long to be expected that an invariant (see [Stev] and [Ji-Jiang], [Jiang-Wang] and [Jiang1]) , we call it $Inv^0(A)$ (see the introduction), consisting of scaled ordered total $K$-group $(\underline{K}(A), \underline{K}(A)^{+},ΣA)_Λ$ (used in the real rank zero case), the tracial state space $T(pAp)$ of cutting down algebra $pAp$ as part of Elliott invariant of $pAp$ (for each $[p]\inΣA$) with a certain compatibility, is the complete invariant for certain well behaved class of $C^*$-algebras with the ideal property (e.g., $AH$ algebras with no dimension growth). In this paper, we will construct two non isomorphic $A\mathbb{T}$ algebras $A$ and $B$ with the ideal property such that $Inv^0(A)\cong Inv^0(B)$. The invariant to differentiate the two algebras is the Hausdorffifized algebraic $K_1$-groups $U(pAp)/\overline{DU(pAp)}$ (for each $[p]\inΣA$) with a certain compatibility condition. It will be proved in [GJL] that, adding this new ingredients, the invariant will become the complete invariant for $AH$ algebras (of no dimension growth) with the ideal property.

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arXiv2018

$C^*$ exponential length of commutators unitaries in $AH$ algebras

For each unital $C^*$-algebra $A$, we denote $cel_{CU}(A)=\sup\{cel(u):u\in CU(A)\}$, where $cel(u)$ is the exponential length of $u$ and $CU(A)$ is the closure of the commutator subgroup of $U_0(A)$. In this paper, we prove that $cel_{CU}(A)=2π$ provided that $A$ is an $AH$ algebras with slow dimension growth whose real rank is not zero. On the other hand, we prove that $cel_{CU}(A)\leq 2π$ when $A$ is an $AH$ algebra with ideal property and of no dimension growth (if we further assume $A$ is not of real rank zero, we have $cel_{CU}(A)= 2π$).

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arXiv2017

A Reduction theorem for $AH$ algebras with ideal property

Let $A$ be an $AH$ algebra, that is, $A$ is the inductive limit $C^{*}$-algebra of $$A_{1}\xrightarrow{ϕ_{1,2}}A_{2}\xrightarrow{ϕ_{2,3}}A_{3}\longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots$$ with $A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $X_{n,i}$ are compact metric spaces, $t_{n}$ and $[n,i]$ are positive integers, and $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that $\sup_{n,i}dim(X_{n,i})<+\infty$. In this article, we prove that $A$ can be written as the inductive limit of $$B_{1}\longrightarrow B_{2}\longrightarrow\cdots\longrightarrow B_{n}\longrightarrow\cdots,$$ where $B_{n}=\bigoplus_{i=1}^{s_{n}}Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, where $Y_{n,i}$ are $\{pt\}$, $[0,1]$,$ S^{1}$,$ T_{II, k},$ $T_{III, k}$ and $S^{2}$ (all of them are connected simplicial complexes of dimension at most three), $s_{n}$ and $\{n,i\}$ are positive integers and $Q_{n,i}\in M_{\{n,i\}}(C(Y_{n,i}))$ are projections. This theorem unifies and generalizes the reduction theorem for real rank zero $AH$ algebras due to Dadarlat and Gong ([D], [G3] and [DG]) and the reduction theorem for simple $AH$ algebras due to Gong (see [G4]).

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