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Chunlan Jiang - One of the best experts on this subject based on the ideXlab platform.
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Inductive Limit of direct sums of simple TAI algebras
Journal of Topology and Analysis, 2019Co-Authors: Bo Cui, Chunlan JiangAbstract:An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive Limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], respectively) is an Inductive Limit [Formula: see text], where each [Formula: see text] is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive Limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404], the second author classified all ATAI algebras by an invariant consisting orderd total [Formula: see text]-theory and tracial state spaces of cut down algebras under an extra restriction that all element in [Formula: see text] are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic [Formula: see text]-group as an addition to the invariant used in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive Limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404]. The theorem is proved by reducing the class to the classification theorem of [Formula: see text] algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of Inductive Limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive Limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (see Corollary 4.3).
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on the Inductive Limit of direct sums of simple tai algebras
arXiv: Operator Algebras, 2019Co-Authors: Bo Cui, Chunlan JiangAbstract:An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an Inductive Limit $\lim\Limits_{n\rightarrow\infty}(A_{n}=\bigoplus\Limits_{i=1}A_{n}^{i},\phi_{nm})$, 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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a classification of Inductive Limit c algebras with ideal property
arXiv: Operator Algebras, 2016Co-Authors: Guihua Gong, Chunlan JiangAbstract:Let $A$ be an $AH$ algebra $A=\lim\Limits_{n\to \infty}(A_{n}=\bigoplus\Limits_{i=1}\Limits^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, \phi_{n,m})$, 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 $(\underline{K}(A), \underline{K}(A)_{+},\Sigma A)$ (as already used in real rank zero case in [DG]), for each $[p]\in\Sigma A$, 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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A Complete Classification of AI Algebras with the Ideal Property
Canadian Journal of Mathematics, 2011Co-Authors: Chunlan JiangAbstract:Abstract Let A be an AI algebra; that is, A is the C*-algebra Inductive Limit of a sequencewhere are [0, 1], kn, and [n, i] are positive integers. 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 give a complete classification of AI algebras with the ideal property.
P. J. Stacey - One of the best experts on this subject based on the ideXlab platform.
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an Inductive Limit model for the k theory of the generator interchanging antiautomorphism of an irrational rotation algebra
Canadian Mathematical Bulletin, 2003Co-Authors: P. J. StaceyAbstract:Let Abe the universal C � -algebra generated by two unitaries U, V satisfying VU = e 2�iUV and letbe the antiautomorphism of Ainterchanging U and V. The K-theory of R� = {a ∈ A� : �(a) = a � } is computed. Whenis irrational, an Inductive Limit of algebras of the form Mq(C(T)) ⊕ Mq'(R) ⊕ Mq(R) is constructed which has complexification Aand the same K-theory as R�.
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Inductive Limit decompositions of real structures in irrational rotation algebras
Indiana University Mathematics Journal, 2002Co-Authors: P. J. StaceyAbstract:An irrational rotation C*-algebra can be written as a direct Limit of algebras, each of which is the direct sum of two matrix algebras over the algebra of continuous functions on the circle. The invariance of such decompositions under involutory antiautomorphisms is investigated, and a complete answer obtained for toral antiautomorphisms.
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Inductive Limit Toral Automorphisms of Irrational Rotation Algebras
Canadian Mathematical Bulletin, 2001Co-Authors: P. J. StaceyAbstract:Irrational rotation C�-algebras have an Inductive Limit decomposition in terms of matrix algebras over the space of continuous functions on the circle and this decomposition can be chosen to be invariant under the flip automorphism. It is shown that the flip is essentially the only toral automorphism with this property.
Qiu Jing-hui - One of the best experts on this subject based on the ideXlab platform.
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WEAKLY COUNTABLE COMPACT REGULARITY OF Inductive LimitS OF WEBBED SPACES
Journal of Mathematics, 2009Co-Authors: Qiu Jing-huiAbstract:In this article, we investigate the regularity of Inductive Limits with respect to weak topologies. By using the method of Banach disks and the localization theorem for strictly webbed spaces, we prove that if an Inductive Limit of strictly webbed spaces satisfies condition (Q0), then, it is convex weakly countably compact regular.
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The categories and the Inductive Limit of Lipschitz function's families
Journal of Suzhou University, 2002Co-Authors: Qiu Jing-huiAbstract:In this paper,by using the theory of FH spaces,we investigate the categories and the Inductive Limit of Lipschitz function's families which define on balls in Banach spaces.
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Sequential retractivities and regularity on Inductive Limits
Czechoslovak Mathematical Journal, 2000Co-Authors: Qiu Jing-huiAbstract:In this paper we prove the following result: an Inductive Limit (E, t) = ind(En, tn) is regular if and only if for each Mackey null sequence (xk) in (E, t) there exists \(n = n\left( {x_k } \right) \in \mathbb{N}\) such that (xk) is contained and bounded in (En, tn). From this we obtain a number of equivalent descriptions of regularity.
Jan Kučera - One of the best experts on this subject based on the ideXlab platform.
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Sequentially Complete Inductive Limits and Regularity
Czechoslovak Mathematical Journal, 2004Co-Authors: Claudia Gomez-wulschner, Jan KučeraAbstract:A notion of an almost regular Inductive Limits is introduced. Every sequentially complete Inductive Limit of arbitrary locally convex spaces is almost regular.
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Sequential completeness and regularity of Inductive Limits of webbed spaces
Czechoslovak Mathematical Journal, 2002Co-Authors: Carlos Bosch, Jan KučeraAbstract:Any Inductive Limit of bornivorously webbed spaces is sequentially complete iff it is regular.
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SEQUENTIAL COMPLETENESS OF Inductive LimitS
International Journal of Mathematics and Mathematical Sciences, 2000Co-Authors: Claudia Gómez, Jan KučeraAbstract:A regular Inductive Limit of sequentially complete spaces is sequentially complete. For the converse of this theorem we have a weaker result: if ind En is sequentially complete Inductive Limit, and each constituent space En is closed in ind En, then ind En is α-regular.
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Regularity of conservative Inductive Limits
International Journal of Mathematics and Mathematical Sciences, 1999Co-Authors: Jan KučeraAbstract:A sequentially complete Inductive Limit of Frechet spaces is regular, see [3]. With a minor modification, this property can be extended to Inductive Limits of arbitrary locally convex spaces under an additional assumption of conservativeness.
Guihua Gong - One of the best experts on this subject based on the ideXlab platform.
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a classification of Inductive Limit c algebras with ideal property
arXiv: Operator Algebras, 2016Co-Authors: Guihua Gong, Chunlan JiangAbstract:Let $A$ be an $AH$ algebra $A=\lim\Limits_{n\to \infty}(A_{n}=\bigoplus\Limits_{i=1}\Limits^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, \phi_{n,m})$, 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 $(\underline{K}(A), \underline{K}(A)_{+},\Sigma A)$ (as already used in real rank zero case in [DG]), for each $[p]\in\Sigma A$, 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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on the classification of simple Inductive Limit c algebras ii the isomorphism theorem
Inventiones Mathematicae, 2007Co-Authors: George A. Elliott, Guihua Gong, Liangqing LiAbstract:In this article, it is proved that the invariant consisting of the scaled ordered K-group and the space of tracial states, together with the natural pairing between them, is a complete invariant for the class of unital simple C *-algebras which can be expressed as the Inductive Limit of a sequence $$A_1\to A_2\to\cdots\to A_n\to\cdots$$ 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 is a compact metrizable space and P n,i is a projection in M [n,i](C(X n,i )) for each n and i, and the spaces X n,i are of uniformly bounded finite dimension. Note that the C *-algebras in the present class are not assumed to be of real rank zero, as they were in [EG2].
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injectivity of the connecting maps in ah Inductive Limit systems
Canadian Mathematical Bulletin, 2005Co-Authors: George A. Elliott, Guihua GongAbstract:Let A be the Inductive Limit of a system A1 xrightarrow(phi1,2) A2 xrightarrow(phi2,3) A3 longrightarrow ... with An = bigoplusi=1tn Pn,i M[n,i](C(Xn,i))Pn,i, where Xn,i is a finite simplicial complex, and Pn,i is a projection in M[n,i](C(Xn,i)). In this paper, we will prove that A can be written as another Inductive Limit B1 xrightarrow(psi1,2) B2 xrightarrow(psi2,3) B3 longrightarrow ... with Bn = bigoplusi=1sn Qn,i M{n,i}(C(Yn,i)) Qn,i, where Yn,i is a finite simplicial complex, and Qn,i is a projection in M{n,i} (C(Yn,i)), with the extra condition that all the maps psin,n+1 are injective. (The result is trivial if one allows the spaces Yn,i to be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras. The special case that the spaces Xn,i are graphs is due to the third named author.
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on the classification of simple Inductive Limit c algebras i the reduction theorem
Documenta Mathematica, 2002Co-Authors: Guihua GongAbstract:Suppose that formula math. is a simple C*-algebra, where X n,i are compact metrizable spaces of uniformly bounded dimensions (this restriction can be relaxed to a condition of very slow dimension growth). It is proved in this article that A can be written as an Inductive Limit of direct sums of matrix algebras over certain special 3-dimensional spaces. As a consequence it is shown that this class of Inductive Limit C*-algebras is classified by the Elliott invariant - consisting of the ordered K-group and the tracial state space - in a subsequent paper joint with G. Elliott and L. Li (Part II of this series). (Note that the C*-algebras in this class do not enjoy the real rank zero property.).
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ON Inductive LimitS OF MATRIX ALGEBRAS OVER THE TWO-TORUS
American Journal of Mathematics, 1996Co-Authors: George A. Elliott, Guihua GongAbstract:It will be shown in this paper that certain real rank zero C*-algebras which are Inductive Limits of C*-algebras of the form ⊕ i M k i (C([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /])) can be expressed as Inductive Limits of C*-algebras of the form ⊕ i M k i (C( S 1 )). In particular, if both A and B are of real rank zero and are Inductive Limits of C*-algebras of the form ⊕ i M k i (C( S 1 )), then also A ⊗ B is an Inductive Limit of C*-algebras of the form ⊕ i M k i (C( S 1 )). (Hence, A ⊗ B can be classified by its K-theory.) This is a key step in the general classification theory of Inductive Limit C*-algebras.
