We prove a classification theorem for simple direct limits of what we
call even Cuntz-circle algebras: finite direct sums of algebras of the form
C(X)®Mn®Om, where m is even, Om is the Cuntz algebra (first introduced
in [Cul]), and X is either a point, a compact interval, or the circle S
. The
unital version of our main theorem is:
T h e o r e m A. (Theorem 5.4) Let A = limA^ and B = limi?*. be simple
separable unital C*-algebras, which are direct limits of even Cuntz-circle
algebras. Assume that the homomorphisms Ak A and Bk » B are
"approximately absorbing" (defined in Section 1). Then A = B if and only
if (Ko{A),[lA]J1(A)) ^ (K0(B),[lB],Ki(B)). In particular, if there are
isomorphisms a
: Ko(A) KQ(B) and a\ : K\(A) K\(B) such that
^OQIA] )
then A is isomorphic to B.
As a corollary, we obtain:
T h e o r e m B . (Corollary 5.12) Let 6\ and 92 be irrational numbers, and
let AQX and AQ2 be the corresponding irrational rotation algebras. Then for
any even m, we have Aex ® Om = l^2 ® Om.
This contrasts with the fact, due to Rieffel [Rf] and Pimsner and Voicu-
lescu [PV1] that A$1 = A^2 only when ^i = ±^
(mod Z). (Theorem B was
already known for m = 2 [Ln3], and remains unknown for odd m.) More
Received by the editor April 12, 1994
This research is partially supported by NSF grants DMS 93-01082 (H. Lin) and
DMS 91-06285 (N. C. Phillips). The results of this article were announced at the
January 1994 meeting of the American Mathematical Society.
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