We define a tracial analog of the notion called sequentially split $$^*$$ -homomorphism between $$C^*$$ -algebras due to Barlak and Szabó show that several important approximation properties related classification theory pass from target algebra domain algebra. Then, we this framework arises Rokhlin finite group action an inclusion unital -algebras.

Let $A$ be a unital simple $A\mathbb{T}$-algebra of real rank zero. Given an order two automorphism $h: K\_1(A)\to K\_1(A)$, we show that there is $\alpha$: $A\to A$ such $\alpha\_{\*0}=\mathrm {id}$, $\alpha\_{1}=h$ and the action $\mathbb{Z}\_2$ generated by $\alpha$ has tracial Rokhlin property. Consequently, $C^(A,\mathbb{Z}\_2,\alpha)$ AH-algebra with no dimension growth, Thus our main res...

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: • AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. • Simple unital AH algebras with slow dimension growth and real rank zero. • C*-algebras with real rank ...

We describe some of the forms of freeness of group actions on noncommutative C*-algebras that have been used, with emphasis on actions of finite groups. We give some indications of their strengths, weaknesses, applications, and relationships to each other. The properties discussed include the Rokhlin property, K-theoretic freeness, the tracial Rokhlin property, pointwise outerness, saturation, ...

Let A be a separable, unital, simple, Z-stable, nuclear C?-algebra, and let ?:G?Aut(A) an action of discrete, countable, amenable group. Suppose that the orbits G on T(A) are finite their cardinality is bounded. We show following equivalent: ? strongly outer; ??idZ has weak tracial Rokhlin property. dimension (in fact, at most 2). If ?eT(A) furthermore compact, covering dimension, orbit space ?...

Let A be a unital simple C∗-algebra with tracial rank zero and X be a compact metric space. Suppose that h1, h2 : C(X) → A are two unital monomorphisms. We show that h1 and h2 are approximately unitarily equivalent if and only if [h1] = [h2] in KL(C(X), A) and τ ◦ h1(f) = τ ◦ h2(f) for every f ∈ C(X) and every trace τ of A.Adopting a theorem of Tomiyama, we introduce a notion of approximate con...

Let A be a unital separable C *-algebra, and D a K1-injective strongly self-absorbing C *-algebra. We show that if A is D-absorbing, then the crossed product of A by a compact second countable group or by Z or by R is D-absorbing as well, assuming the action satisfying a Rokhlin property. In the case of a compact Rokhlin action we prove a similar statement about approximate divisibility.

Let Γ be the convex set consisting of all states φ on the tensor product B ⊗ B of the algebra B = Mn(C) of all n × n matrices over the complex numbers C with the property that the restrictions φ B⊗I and φ I⊗B are the unique tracial states on B ⊗ I and I ⊗ B . We find necessary and sufficient conditions for such a state, called a marginal tracial state, to be extremal in Γ . We also give a chara...

and Applied Analysis 3 Lemma 2.2 see 5, Theorem 3.3 , 6, Theorem 3.3 . Let A be a simple unital C∗-algebra. If TRR A 0, then RR A 0. If Tsr A 1 and has the (SP)-property, then tsr A 1. Definition 2.3 see 13, Definition 1.2 . Let A be an infinite dimensional finite simple separable unital C∗-algebra, and let α : G → Aut A be an action of a finite group G on A. We say that α has the tracial Rokhl...