Math. Proc. Camb. Phil. Soc. (1999), 126, 555 Printed in the United Kingdom c  1999 Cambridge Philosophical Society 555 A simple local multiplier algebra By PERE ARA† Departament de Matem` atiques, Universitat Aut` onoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain e-mail: para@mat.uab.es and MARTIN MATHIEU† Department of Pure Mathematics, The Queen’s University of Belfast, Belfast, BT71NN, Northern Ireland e-mail: mm@qub.ac.uk (Received 24 October 1997; revised 13 February 1998) Abstract We present a class of approximately finite dimensional C*-algebras whose local multiplier algebras are simple with real rank zero and stable rank one. The local multiplier algebra of a C*-algebra A, that is, the direct limit of the mul- tiplier algebras of the closed essential ideals of A, has proved to be a useful device in operator theory on C*-algebras [1, 2, 4, 11, 12, 15]. For example, if δ a is an inner derivation on A implemented by an element a in the multiplier algebra M (A) of A, there is always a local multiplier a ′ of A such that δ a = δ a ′ and ‖δ a ‖ =2 ‖a ′ ‖, as a consequence of the results in [10] and [14]. These applications stimulated the inves- tigation of the structure of the local multiplier algebra M loc (A), for a comprehensive account see [3]. Built from multiplier algebras of (closed essential) ideals of A, M loc (A) shares some of the properties of M (A); however, it incorporates the details of the ideal structure of A. To illustrate this, let us recall that a C*-algebra A is prime if and only if M loc (A) has trivial centre. A natural question on the ideal structure of M loc (A) itself is the following. Under what conditions will M loc (A) be simple? Whereas M (A) can only be simple in the trivial case, i.e. if A is simple and unital, it may be possible for M loc (A) to be simple without being equal to A. In the paper at hand we shall present a class of C*-algebras where this occurs. A necessary condition is that A must be prime; in fact, our class consists of unital (separable) AF-algebras (which are therefore primitive). Since unital AF-algebras are completely classified by their scaled ordered K 0 -groups, the main point in our approach will be the construction † This paper was written during a Research in Pairs stay of the authors at the Mathematical Research Institute in Oberwolfach generously supported by the Volkswagen-Stiftung. The first- named author was also supported in part by DGICYT grant PB95-0626 as well as the Commissionat per Universitats i Recerca de la Generalitat de Catalunya. The second-named author was a Marie Curie Fellow under the Training and Mobility of Researchers scheme of the European Commission. www.DownloadPaper.ir www.DownloadPaper.ir