Nonstable K-theory for Z-stable C-algebras

Let Z denote the simple limit of prime dimension drop algebras that has a unique tracial state (cf. Jiang and Su [11]). Let A 6= 0 be a unital C∗-algebra with A ∼= A ⊗ Z. Then the homotopy groups of the group U(A) of unitaries in A are stable invariants, namely, πi(U(A)) ∼= Ki−1(A) for all integer i ≥ 0. Furthermore, A has cancellation for full projections, and satisfies the comparability question for full projections. Analogous results hold for non-unital Z-stable C∗-algebras. 0. Introduction and summary of results Let Z denote the only simple limit of prime dimension drop algebras that has a unique tracial state (cf. Jiang and Su [11]). A C-algebra A is called Z-stable, if A ∼= A ⊗ Z. In this note, we study nonstable K-theory for Z-stable C-algebras. Our motivations first came from works on approximately divisible C-algebras (cf. Blackadar, Kumjian and Rordam [4] and the references therein). At first glance, these two classes of algebras might seem quite unrelated. For example, the algebra Z itself is known to be Z-stable ([11]). Since Z has no non-trivial projections, it is certainly not approximately divisible. It is quite easy to check that some properties (related to projections) of approximately divisible algebras (cf. Theorem 1.4(b), (e), (f) in [4]) fail on Z. On the other hand, some very interesting approximately divisible C-algebras are known to be Z-stable (cf. Theorem 5 of [11]). In fact, to this date we know of no obstructions to Z-stability for approximately divisible (nuclear) C-algebras. Furthermore, non-zero Z-stable C-algebras are, in a certain sense (cf. Remark 1.4), “fibrewise approximately divisible”, and certain properties of approximately divisible C-algebras do persist in the class of Z-stable algebras (cf. Theorem 3 of Gong, Jiang and Su [9]). In this note, we establish the following results (compare Theorem 1.4(d), Propositions 3.10, and Proposition 3.11 of [4]): Theorem 1. Let A be a unital Z-stable C-algebra. Then: (a) A has cancellation for full projections: If p and q are two full projections in A, and [p] = [q] ∈ K0(A), then e and f are (Murray-von Neumann) equivalent. (b) A satisfies the comparability question for full projections: If p and q are two full projections in A, and τ(p) < τ(q) for all quasi-traces τ on A, then p is equivalent to a subprojection of q. Theorem 2. Let A be a unital Z-stable C-algebra. Then the natural map μ : U(A)/U0(A) → K1(A) is an isomorphism, where U(A) denotes the unitary group of A and U0(A) the connected component of the identity. Date: Nov. 12, 1997. 1991 Mathematics Subject Classification. Primary:46L80; Secondary: 46L05, 19K14, 55Q52.