Simple C ∗ -Algebras and Subgroups of Q

Abstract

A special case of a conjecture of R. Douglas is solved by an elementary argument using K0-theory. Let F be a subgroup of the additive reals R and let F+ = {x E F: x > O}. Douglas [2] defines a one-parameter semigroup of isometries to be a homomorphism x ~-4 V of F+ into the set of isometries on some Hilbert space H (i.e. x V = VyV? for x ,y E F+ and V0 = 1 ). Denoting by AF(VX) the C*-algebra generated by all VT (x E F+) and calling the map x 4 V nonunitary if no Tx is unitary except VJ = 1 , he shows that if x 4 VT and x W are nonunitary one-parameter semigroups of isometries on F then the algebras AF(TV) and AF(KV) are canonically isomorphic. Thus one can speak of Ar (isomorphic to AF( Vy) ) and of its commutator ideal CrF. Douglas shows that CF is simple, and that if Iand F2 are subgroups of R, then Ar and ArF are *-isomorphic iff ri and I2 are isomorphic as ordered groups. He obtains other interesting results on these algebras and conjectures that CF and C-, are *-isomorphic implies that Iand I2 are isomorphic as ordered groups. In this paper we show that CF is an AF-algebra for F a subgroup of Q (the additive rationals) and that in this case we have KO(CF) = F where K0( ) denotes the K0-group of Cr. (For a good account of K0-theory see Goodearl [3].) From this we deduce that Douglas' conjecture is true for subgroups of Q. Douglas was led to investigating these algebras Ar in the context of a generalized Toeplitz theory. The author has shown they satisfy a certain universal property which can facilitate their analysis, and he has generalized them by associating with every ordered group G a C*-algebra which reflects both order and algebra properties of G. The results presented here are part of an ongoing investigation of this more general theory, of which the author intends to give a fuller account in a forthcoming paper. Let H be a separable infinite-dimensional Hilbert space, and let U be the unilateral shift on H. We denote by C the C*-subalgebra of B(H) generated by U, and by K the commutator ideal of C. (The commutator ideal of a Received by the editors August 31, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L80. (? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page