Let A be a unital separable amenable C∗-algebra and C be a unital C∗-algebra with certain infinite property. We show that two full monomorphisms h1, h2 : A → C are approximately unitarily equivalent if and only if [h1] = [h2] in KL(A,C). Let B be a non-unital but σ-unital C ∗-algebra for which M(B)/B has the certain infinite property. We prove that two full essential extensions are approximatel...

We define natural A ∞ -transformations and construct A ∞ -category of A ∞ -functors. The notion of non-strict units in an A ∞ -category is introduced. The 2-category of (unital) A ∞ -categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta6...

We prove that the infinite tensor power of a unital separable C∗-algebra absorbs the Jiang-Su algebra Z tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C∗-algebras without characters known to lie outside it. This category moreover contains the ...

Let A and C be two unital simple C*-algebras with tracial rank zero. Suppose that C is amenable and satisfies the Universal Coefficient Theorem. Denote by KKe(C,A) ++ the set of those κ for which κ(K0(C)+ \ {0}) ⊂ K0(A)+ \ {0} and κ([1C ]) = [1A]. Suppose that κ ∈ KKe(C,A) . We show that there is a unital monomorphism φ : C → A such that [φ] = κ. Suppose that C is a unital AH-algebra and λ : T(...

Abstract. When a unital C*-algebra A has topological stable rank one (write tsr(A) = 1), we know that tsr(pAp) ≤ 1 for a non-zero projection p ∈ A. When, however, tsr(A) ≥ 2, it is generally faluse. We prove that if a unital C*algebra A has a simple unital C*-subalgebra D of A with common unit such that D has Property (SP) and supp∈P (D) tsr(pAp) < ∞, then tsr(A) ≤ 2. As an application let A be...

We prove that under a mild hypothesis, the infinite tensor power of a unital separable C∗-algebra absorbs the Jiang–Su algebra Z tensorially. Combining this result with a recent theorem of Winter, we complete Elliott’s classification program for strongly self-absorbing ASH algebras. We also give a succinct universal property for Z in an ambient category so large that there are no unital separab...

Let D and A be unital and separable C∗-algebras; let D be strongly selfabsorbing. It is known that any two unital ∗-homomorphisms from D to A ⊗ D are approximately unitarily equivalent. We show that, if D is also K1-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of D is asymptotically inner. Moreover, the space of automorphi...

LetX be a compact metric space and let Λ be a Z (k ≥ 1) action onX.We give a solution to a version of Voiculescu’s problem of AF-embedding: The crossed product C(X) ⋊Λ Z k can be embedded into a unital simple AF-algebra if and only if X admits a strictly positive Λ-invariant Borel probability measure. Let C be a unital AH-algebra, let G be a finitely generated abelian group and let Λ : G → Aut(...

The set SectA of all unitary equivalence classes of unital ∗-endomorphisms of a unital C∗-algebra A is called the sector of A. We show that there is an exotic algebraic structure on SectA when A includes a Cuntz algebra as a C∗-subalgebra with common unit. Next we explain that the set BSpecA of all unitary equivalence classes of unital ∗-representations of A is a right module of SectA. An essen...

We present a new construction of non-classical unitals from a classical unital U in PG(2, q2). The resulting non-classical unitals are B–M unitals. The idea is to find a non-standard model Π of PG(2, q2) with the following three properties: (i) points of Π are those of PG(2, q2); (ii) lines of Π are certain lines and conics of PG(2, q2); (iii) the points in U form a non-classical B–M unital in ...