Approximate unitary equivalence in simple $C^{*}$-algebras of tracial rank one

Abstract

Let C be a unital AH-algebra and let A be a unital separable simple C * -algebra with tracial rank no more than one. Suppose that φ, ψ : C → A are two unital monomorphisms. With some restriction on C, we show that φ and ψ are approximately unitarily equivalent if and only ifwhere φ ‡ and ψ ‡ are homomorphisms from U (C)/CU (C) → U (A)/CU (A) induced by φ and ψ, respectively, and where CU (C) and CU (A) are closures of the subgroup generated by commutators of the unitary groups of C and B.A more practical but approximate version of the above is also presented.

The Tracial Topological Rank of Extensions of $$C^*$$ C ∗ -Algebras

“…Since By (5), without loss of generality, we may assume π( p F p ) ⊆ B 2 . Since the extension 0 → p J p → A → π(A ) → 0 is quasidiagonal, by (13), (14), (19) and Theorem 3.2, there exist a projection q and a C * -subalgebra A 1 ∈ I (1) of A with 1 A 1 = q, π(q) =q and π(A 1 ) = B 2 , and a projection r ∈ ( p − q)J ( p − q) such that (20) qx − xq < 8ξ and r x − xr < 24ξ, ∀ x ∈ q Fq; (21) qxq ∈ 2ξ C and (q + r )x(q + r ) ∈ 18ξ A 1 + r Jr, ∀ x ∈ q Fq; (22) (n + 1)[ f α 5 α 6 (( p − q − r )a( p − q − r ))] ≤ [ f d 1 d 2 (qaq)]. Let A 2 = A 1 + r Jr.…”

Homotopy of unitaries in simpleC∗-algebras with tracial rank one

“…It has been shown in [22] that any k-dimensional torus has the property (H). So do those finite CW complexes X with torsion free K 0 (C(X)) and torsion K 1 (C(X)), any finite CW complexes with form Y × T where Y is contractive and all one-dimensional finite CW complexes.…”

“…) be as required by Theorem 10.8 of [22] corresponding to /16 (in place of ), F 1 and 1 (in place of ) above.…”

“…In a subsequent paper [23], we use the main homotopy result (Theorem 8.4) of this paper and the results in [22] to establish a K-theoretical necessary and sufficient condition for homomorphisms from unital simple AH-algebras into a unital separable simple C * -algebra with tracial rank no more than one to be asymptotically unitarily equivalent which, in turn, combining with a result of W. Winter, provides a classification theorem for a class of unital separable simple amenable C * -algebras which properly contains all unital separable simple amenable C * -algebras with tracial rank no more than one which satisfy the UCT as well as some projectionless C * -algebras such as the Jiang-Su algebra.…”

Classification of homomorphisms into simpleZ-stableC⁎-algebras

“…Nielsen and K. Thomsen [25] obtained the analogous result for general AT algebras. H. Lin [17,20] classified unital homomorphisms from AH algebras into simple separable C * -algebras of tracial rank no more than one. Classification up to asymptotic unitary equivalence is also studied in [27,13,18].…”