Let （E,ξ）=ind（En,ξn） be an inductive limit of a sequence （En,ξn）n∈N of locally convex spaces and let every step （En,ξn） be endowed with a partial order by a pointed convex （solid） cone Sn. In the framew...Let （E,ξ）=ind（En,ξn） be an inductive limit of a sequence （En,ξn）n∈N of locally convex spaces and let every step （En,ξn） be endowed with a partial order by a pointed convex （solid） cone Sn. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence （Sn）n∈N of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.展开更多
This paper is a survey on the recent work of the authors and their col-laborators on the Classification of Inductive Limit C*-algebras. Some examples are presented to explain several important ideas.
Let (E,ζ)= indlim (E n ,ζ n ) be an inductive limit of locally convex spaces. We say that ( DST ) holds if each bounded set in (E,ζ) is contained and bounded in some (E n ,ζ n ). We introdu...Let (E,ζ)= indlim (E n ,ζ n ) be an inductive limit of locally convex spaces. We say that ( DST ) holds if each bounded set in (E,ζ) is contained and bounded in some (E n ,ζ n ). We introduce a property which is weaker than fast completeness, quasi-fast completeness, and prove that for inductive limits of strictly webbed spaces, quasi-fast completeness implies that ( DST ). By using De Wilde’s theory on webbed spaces,we also give some other conditions for ( DST ). These results improve relevant earlier results.展开更多
We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C^(*)-algebra. We also show that if a unital C^(*)-algebra can be approximately embedded into some tensorially self abs...We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C^(*)-algebra. We also show that if a unital C^(*)-algebra can be approximately embedded into some tensorially self absorbing C^(*)-algebra C(e.g., uniformly hyperfinite(UHF)-algebras of infinite type, the Cuntz algebra O_(2)),then we can construct a simple separable unital generalized inductive limit. When C is simple and infinite(resp.properly infinite), the construction is also infinite(resp. properly infinite). When C is simple and approximately divisible, the construction is also approximately divisible. When C is a UHF-algebra and the connecting maps satisfy a trace condition, the construction has tracial rank zero.展开更多
The paper is devoted to the study of generalized inductive limit of C＊-algebras with coherent maps being completely positive contractions of order zero： the nuclear dimension of generalized inductive limit of C＊-al...The paper is devoted to the study of generalized inductive limit of C＊-algebras with coherent maps being completely positive contractions of order zero： the nuclear dimension of generalized inductive limit of C＊-algebras with finite nuclear dimension is finite; the generalized inductive limits of C＊-algebras with the α-comparison property also have the s-comparison property.展开更多
In this article,we introduce and study the class of approximately Artinian(Noetherian)C^(*)-algebras,called AR-algebras(AN-algebras),which is a simultaneous generalization of Artinian(Noetherian)C*-algebras and AF-alg...In this article,we introduce and study the class of approximately Artinian(Noetherian)C^(*)-algebras,called AR-algebras(AN-algebras),which is a simultaneous generalization of Artinian(Noetherian)C*-algebras and AF-algebras.We study properties such as the ideal property and topological dimension zero for them.In particular,we show that a faithful AR or AN algebra is strongly purely infinite iff it is purely infinite iff it is weakly purely infinite.This extends the Kirchberg's O_(∞)-absorption theorem,and implies that a weakly purely infinite C^(*)-algebra is Noetherian iff every its ideal has a full projection.展开更多
Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special ...Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, KSthe （LF）-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems.展开更多
Weakly (sequentially) compactly regular inductive limits and convex weakly (sequentially) compactly regular inductive limits are investigated. (LF)-spaces satisfying Retakh's condition (M0) are convex weakly (sequ...Weakly (sequentially) compactly regular inductive limits and convex weakly (sequentially) compactly regular inductive limits are investigated. (LF)-spaces satisfying Retakh's condition (M0) are convex weakly (sequentially) compactly regular but need not be weakly (sequentially) compactly regular. For countable inductive limits of weakly sequentially complete Frechet spaces, Retakh's condition (M0) implies weakly (sequentially) compact regularity. Particularly for Kothe (LF)-sequence spaces Ep(1 ≤ p < ∞), Retakh's condition (M0) is equivalent to weakly (sequentially) compact regularity. For those spaces, the characterizations of weakly (sequentially) compact regularity are given by using the related results of Vogt.展开更多
We define a generalization of Mackey first countability and prove that it is equivalent to being docile. A consequence of the main result is to give a partial affirmative answer to an old question of Mackey regarding ...We define a generalization of Mackey first countability and prove that it is equivalent to being docile. A consequence of the main result is to give a partial affirmative answer to an old question of Mackey regarding arbitrary quotients of Mackey first countable spaces. Some applications of the main result to spaces such as inductive limits are also given.展开更多
基金supported by the National Natural Science Foundation of China(10871141)
文摘Let （E,ξ）=ind（En,ξn） be an inductive limit of a sequence （En,ξn）n∈N of locally convex spaces and let every step （En,ξn） be endowed with a partial order by a pointed convex （solid） cone Sn. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence （Sn）n∈N of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.
基金Both authors are supported by NSF grant DMS9970840 This material is also based uponwork supported by,the U.S. Army Research Office under grant number DAADl9-00-1-0152 for both authors.
文摘This paper is a survey on the recent work of the authors and their col-laborators on the Classification of Inductive Limit C*-algebras. Some examples are presented to explain several important ideas.
文摘Let (E,ζ)= indlim (E n ,ζ n ) be an inductive limit of locally convex spaces. We say that ( DST ) holds if each bounded set in (E,ζ) is contained and bounded in some (E n ,ζ n ). We introduce a property which is weaker than fast completeness, quasi-fast completeness, and prove that for inductive limits of strictly webbed spaces, quasi-fast completeness implies that ( DST ). By using De Wilde’s theory on webbed spaces,we also give some other conditions for ( DST ). These results improve relevant earlier results.
基金supported by the Research Center for Operator Algebras at East China Normal University which is funded by the Science and Technology Commission of Shanghai Municipality (Grant No.13dz2260400)National Natural Science Foundation of China (Grant No.11531003)+1 种基金Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice (Grant No.1361431)the special fund for the Short-Term Training of Graduate Students from East China Normal University。
文摘We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C^(*)-algebra. We also show that if a unital C^(*)-algebra can be approximately embedded into some tensorially self absorbing C^(*)-algebra C(e.g., uniformly hyperfinite(UHF)-algebras of infinite type, the Cuntz algebra O_(2)),then we can construct a simple separable unital generalized inductive limit. When C is simple and infinite(resp.properly infinite), the construction is also infinite(resp. properly infinite). When C is simple and approximately divisible, the construction is also approximately divisible. When C is a UHF-algebra and the connecting maps satisfy a trace condition, the construction has tracial rank zero.
基金Supported by NSFC(Grant No.11371279)by the Fundamental Research Funds for the Central Universities
文摘The paper is devoted to the study of generalized inductive limit of C＊-algebras with coherent maps being completely positive contractions of order zero： the nuclear dimension of generalized inductive limit of C＊-algebras with finite nuclear dimension is finite; the generalized inductive limits of C＊-algebras with the α-comparison property also have the s-comparison property.
基金supported by grants from INSF(98029498,99013953)partly supported by a grant from IPM(96430215)。
文摘In this article,we introduce and study the class of approximately Artinian(Noetherian)C^(*)-algebras,called AR-algebras(AN-algebras),which is a simultaneous generalization of Artinian(Noetherian)C*-algebras and AF-algebras.We study properties such as the ideal property and topological dimension zero for them.In particular,we show that a faithful AR or AN algebra is strongly purely infinite iff it is purely infinite iff it is weakly purely infinite.This extends the Kirchberg's O_(∞)-absorption theorem,and implies that a weakly purely infinite C^(*)-algebra is Noetherian iff every its ideal has a full projection.
基金Supported by the National Natural Science Foundation of China(10571035,10871141)
文摘Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, KSthe （LF）-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems.
基金Supported by the Natural Science Foundation of the Education Committee of Jiangsu Province (Q1107107)
文摘Weakly (sequentially) compactly regular inductive limits and convex weakly (sequentially) compactly regular inductive limits are investigated. (LF)-spaces satisfying Retakh's condition (M0) are convex weakly (sequentially) compactly regular but need not be weakly (sequentially) compactly regular. For countable inductive limits of weakly sequentially complete Frechet spaces, Retakh's condition (M0) implies weakly (sequentially) compact regularity. Particularly for Kothe (LF)-sequence spaces Ep(1 ≤ p < ∞), Retakh's condition (M0) is equivalent to weakly (sequentially) compact regularity. For those spaces, the characterizations of weakly (sequentially) compact regularity are given by using the related results of Vogt.
文摘We define a generalization of Mackey first countability and prove that it is equivalent to being docile. A consequence of the main result is to give a partial affirmative answer to an old question of Mackey regarding arbitrary quotients of Mackey first countable spaces. Some applications of the main result to spaces such as inductive limits are also given.
