Some Properties of Tracially Quasidiagonal Extensions

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))| < ∈).

Comparison Properties for Asymptotically Tracially Approximation C^(*)-algebras

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 < ∞).

The tracial topological rank of certain C*-algebras

Let 0 →I → A →A/I →0 be a short exact sequence of C^＊-algebras with A unital. Suppose that I has tracial topological rank no more than one and A/I belongs to a class of certain C^＊-algebras. We show that A has trazial topological rank no more than one if the extension is quasidiagonal, and A has the property （P1） if the extension is tracially quasidiagonal.

Tracial Limit of C^＊-algebras

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.

Certain Cuntz Semigroup Properties of Certain Crossed Product C^*-algebras

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.

Simple generalized inductive limits of C^(*)-algebras

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.

Surjective L^(2)-isometries on the Projection Lattice

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.

Classification of Homomorphisms from C(X)to Simple C~*-Algebras of Real Rank Zero

Let A be a unital simple C-algebra of real rank zero,stable rank one,with weakly unperforated K0（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.

The Tracial Rokhlin Property for Automorphisms on Non-simple C~*-Algebras

Let A be a unital AF-algebra （simple or non-simple） and let α be an automorphism of A. Suppose that α has certain Rokhlin property and A is α-simple. Suppose also that there is an integer J ≥ 1 such that J α＊^J0 =idKo（A）. The author proves that A α Z has tracial rank zero.

Distance Between Unitary Orbits of Self-Adjoint Elements in C^(*)-Algebras of Tracial Rank One

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).

Hereditary uniform property Γ

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.