Let A and B be separable C∗-algebras, A unital and B stable. It is shown that there is a natural six-terms exact sequence which relates the group which arises by considering all semi-split extensions of A by B to the group which arises by restricting the attention to unital semi-split extensions of A by B. The six-terms exact sequence is an unpublished result of G. Skandalis. Let A,B be separab...

Let C be a unital AH-algebra and A be a unital simple C∗-algebra with tracial rank zero. It has been shown that two unital monomorphisms φ, ψ : C → A are approximately unitarily equivalent if and only if [φ] = [ψ] in KL(C,A) and τ ◦ φ = τ ◦ ψ for all τ ∈ T (A), where T (A) is the tracial state space of A. In this paper we prove the following: Given κ ∈ KL(C,A) with κ(K0(C)+\{0}) ⊂ K0(A)+\{0} an...

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 $CU(A)$ closure commutator subgroup $U_0(A)$. In this paper, prove that $cel_{CU}(A)=2\pi$ provided $A$ an $AH$ algebras with slow dimension growth whose real rank not zero. On other hand, $cel_{CU}(A)\leq 2\pi$ when algebra ideal property no (if further assu...

If there is an order-preserving group isomorphism from a directed abelian group G with a finitely generated positive cone G+ onto a simplicial group, then G is called a semisimplicial group. By factoring out the torsion subgroup of a unital group having a finite unit interval, one obtains a semisimplicial unital group. We exhibit a representation for the base-normed space associated with a semi...

A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be...

Kadison’s theorem of 1951 describes the unital surjective isometries between unital C*-algebras as the Jordan *-isomorphisms. We propose a nonselfadjoint version of his theorem and discuss the cases in which this is known to be true.

Let C and A be two unital separable amenable simple C-algebras with tracial rank no more than one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(c)ut = φ2(c) for all c ∈ C if and only if [φ1] = [φ2] in KK(C,A), φ ‡ 1 = φ 2 ...

Let C be a unital AH-algebra and let A be a unital separable simple C∗-algebra with tracial rank zero. Suppose that φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(a)ut = φ2(a) for all a ∈ C if and only if [φ1] = [φ2] in KK(C,A), τ ◦ φ1 = τ ◦ φ2 for all τ ∈ T (A) and the rotation map η̃φ1,φ2 associate...

It is shown that the matrix normed structure of a non-unital operator algebra determines that of its unitization. This makes the study of certain unital operator algebras much easier and provides several interesting