Dimension Functions on Simple C *-Algebras

In order to make available for C*-atgebras the results of Goodearl and Handelman [5] on existence and uniqueness of rank functions on regular rings, we associate in the present note with every C*-algebra A an abelian group K*(A). The construction of this group is analogous to the construction of the Grothendieck group Ko(A ) (which recently has been applied successfully to the classification of AF-algebras by G. Elliott [4]) but the group K~(A) itself is in general quite different from Ko(A ), We are mainly interested in the case where A is a simple C*algebra with unit. If A is such an algebra we call A finite, if x*x =a implies xx* =1 (x~A), and we call A stably finite if M,®A (M,=C*-algebra of n× n complex matrices) is finite for all n~ N. Then K](A) is non-trivial if and only if A is stably finite. We note, incidentally, that it was shown in [3, 2.4] that a simple C*-algebra A with unit is stably finite if and only if ~ ® A (grf= C*-algebra of compact operators on a separable infinite-dimensional Hilbert space) contains at least one non-trivial ideal (which can of course not be closed). It is an open problem which is probably difficult to decide if every finite simple C*-algebra with unit is stably finite. However, if one studies traces on A, one may restrict attention to stably finite algebras. In fact, if there is a trace on A, then A is stably finite.