We show that the following properties of the C*-algebras in a class P are inherited by simple unital C*-algebras in the class of asymptotically tracially in P :(1) n-comparison,(2) α-comparison(1 ≤ α < ∞).
Suppose that 0→ I→ A→ A/I→ 0 is a tracially quasidiagonal extension of C*-algebras. In this paper, the authors give two descriptions of the K_0, K_1 index maps which are induced by the above extension and show tha...Suppose that 0→ I→ A→ A/I→ 0 is a tracially quasidiagonal extension of C*-algebras. In this paper, the authors give two descriptions of the K_0, K_1 index maps which are induced by the above extension and show that for any ∈ > 0, any τ in the tracial state space of A/I and any projection p ∈ A/I(any unitary u ∈ A/I), there exists a projection p ∈ A(a unitary u ∈ A) such that |τ(p)-τ(π(p))| < ∈(|τ(u)-τ(π(u))| < ∈).展开更多
The note studies certain distance between unitary orbits.A result about Riesz interpolation property is proved in the first place.Weyl(1912) shows that dist(U(x),U(y))= δ(x,y) for self-adjoint elements in matrixes.Th...The note studies certain distance between unitary orbits.A result about Riesz interpolation property is proved in the first place.Weyl(1912) shows that dist(U(x),U(y))= δ(x,y) for self-adjoint elements in matrixes.The author generalizes the result to C*-algebras of tracial rank one.It is proved that dist(U(x),U(y)) = D_(c)(x,y) in unital AT-algebras and in unital simple C*-algebras of tracial rank one,where x,y are self-adjoint elements and D_(C)(x,y) is a notion generalized from δ(x,y).展开更多
We study the uniform property Γ for separable simple C^(*)-algebras which have quasitraces and may not be exact. We show that a stably finite separable simple C^(*)-algebra A with the strict comparison and uniform pr...We study the uniform property Γ for separable simple C^(*)-algebras which have quasitraces and may not be exact. We show that a stably finite separable simple C^(*)-algebra A with the strict comparison and uniform property Γ has tracial approximate oscillation zero and stable rank one. Moreover in this case,its hereditary C^(*)-subalgebras also have a version of uniform property Γ. If a separable non-elementary simple amenable C^(*)-algebra A with strict comparison has this hereditary uniform property Γ, then A is Z-stable.展开更多
A new limit of C*-algebras, the tracial limit, is introduced in this paper. We show that a separable simple C*-algebra A is a tracial limit of C*-algebras in I^(k) if and only if A has tracial topological rank no more...A new limit of C*-algebras, the tracial limit, is introduced in this paper. We show that a separable simple C*-algebra A is a tracial limit of C*-algebras in I^(k) if and only if A has tracial topological rank no more than k. We present several known results using the notion of tracial limits.展开更多
We show that the following properties of the C^*-algebras in a class Ω are inherited by simple unital C-algebras in the class TAΩ:(1)(m,n)-decomposable,(2) weakly(m,n)-divisible,(3) weak Riesz interpolation.As an ap...We show that the following properties of the C^*-algebras in a class Ω are inherited by simple unital C-algebras in the class TAΩ:(1)(m,n)-decomposable,(2) weakly(m,n)-divisible,(3) weak Riesz interpolation.As an application,let A be an infinite dimensional simple unital C-algebra such that A has one of the above-listed properties.Suppose that α:G→Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property.Then the crossed product C^*-algebra C^*(G,A,α) also has the property under consideration.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
Recently,Gehér and Semrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries.In this pap...Recently,Gehér and Semrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries.In this paper,we study the surjective L^(2)-isometries of the projection lattice of an infinite dimensional Hilbert space and show that every such isometry can also be described by unitaries and antiunitaries.展开更多
Let A be a unital simple C-algebra of real rank zero,stable rank one,with weakly unperforated K<sub>0</sub>（A）and unique normalized quasi-trace τ,and let X be a compact metric space.We show that two mon...Let A be a unital simple C-algebra of real rank zero,stable rank one,with weakly unperforated K<sub>0</sub>（A）and unique normalized quasi-trace τ,and let X be a compact metric space.We show that two monomorphisms Φ,Ψ:C（X）→A are approximately unitarily equivalent if and only if Φ and Ψ induce the same element in KL（C（X）,A）and the two linear functionals τ ο Φ and τ ο Φ are equal.We also show that,with an injectivity condition,an almost multiplicative morphism from C（X） into A with vanishing KK-obstacle is close to a homomorphism.展开更多
基金Supported by the National Natural Sciences Foundation of China (Grant No. 11871375)。
文摘We show that the following properties of the C*-algebras in a class P are inherited by simple unital C*-algebras in the class of asymptotically tracially in P :(1) n-comparison,(2) α-comparison(1 ≤ α < ∞).
基金supported by the National Natural Science Foundation of China(Nos.11871375,11371279,11601339)Zhejiang Provincial Natural Science Foundation of China(No.LY13A010021)
文摘Suppose that 0→ I→ A→ A/I→ 0 is a tracially quasidiagonal extension of C*-algebras. In this paper, the authors give two descriptions of the K_0, K_1 index maps which are induced by the above extension and show that for any ∈ > 0, any τ in the tracial state space of A/I and any projection p ∈ A/I(any unitary u ∈ A/I), there exists a projection p ∈ A(a unitary u ∈ A) such that |τ(p)-τ(π(p))| < ∈(|τ(u)-τ(π(u))| < ∈).
文摘The note studies certain distance between unitary orbits.A result about Riesz interpolation property is proved in the first place.Weyl(1912) shows that dist(U(x),U(y))= δ(x,y) for self-adjoint elements in matrixes.The author generalizes the result to C*-algebras of tracial rank one.It is proved that dist(U(x),U(y)) = D_(c)(x,y) in unital AT-algebras and in unital simple C*-algebras of tracial rank one,where x,y are self-adjoint elements and D_(C)(x,y) is a notion generalized from δ(x,y).
基金supported by National Science Foundation of USA (Grant No.DMS1954600)the Research Center for Operator Algebras in East China Normal University which is supported by Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice,Science and Technology Commission of Shanghai Municipality (Grant No.22DZ2229014)。
文摘We study the uniform property Γ for separable simple C^(*)-algebras which have quasitraces and may not be exact. We show that a stably finite separable simple C^(*)-algebra A with the strict comparison and uniform property Γ has tracial approximate oscillation zero and stable rank one. Moreover in this case,its hereditary C^(*)-subalgebras also have a version of uniform property Γ. If a separable non-elementary simple amenable C^(*)-algebra A with strict comparison has this hereditary uniform property Γ, then A is Z-stable.
文摘A new limit of C*-algebras, the tracial limit, is introduced in this paper. We show that a separable simple C*-algebra A is a tracial limit of C*-algebras in I^(k) if and only if A has tracial topological rank no more than k. We present several known results using the notion of tracial limits.
基金Supported by National Natural Sciences Foundation of China(Grant Nos.11501357 and 11571008)。
文摘We show that the following properties of the C^*-algebras in a class Ω are inherited by simple unital C-algebras in the class TAΩ:(1)(m,n)-decomposable,(2) weakly(m,n)-divisible,(3) weak Riesz interpolation.As an application,let A be an infinite dimensional simple unital C-algebra such that A has one of the above-listed properties.Suppose that α:G→Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property.Then the crossed product C^*-algebra C^*(G,A,α) also has the property under consideration.
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 in part by NFS of China(Grant Nos.11871303,11971463,11671133)supported in part by NFS of China(Grant Nos.11871127,11971463)+2 种基金supported in part by NFS of China(Grant Nos.11871303,11871127,11971463)NSF of Shandong Province(Grant No.ZR2019MA039)Chongqing Science and Technology Commission(Grant No.cstc2019jcyj-msxm X0256)。
文摘Recently,Gehér and Semrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries.In this paper,we study the surjective L^(2)-isometries of the projection lattice of an infinite dimensional Hilbert space and show that every such isometry can also be described by unitaries and antiunitaries.
基金Research partially supported by NSF Grants DMS 93-01082(H.L)and DMS-9401515(G.G)This work was reported by the first named author at West Coast Operator Algebras Seminar(Sept.1995,Eugene,Oregon)
文摘Let A be a unital simple C-algebra of real rank zero,stable rank one,with weakly unperforated K<sub>0</sub>（A）and unique normalized quasi-trace τ,and let X be a compact metric space.We show that two monomorphisms Φ,Ψ:C（X）→A are approximately unitarily equivalent if and only if Φ and Ψ induce the same element in KL（C（X）,A）and the two linear functionals τ ο Φ and τ ο Φ are equal.We also show that,with an injectivity condition,an almost multiplicative morphism from C（X） into A with vanishing KK-obstacle is close to a homomorphism.
