Math. Proc. Camb. Phil. Soc. (1994), 115, 27 27 Printed in Great Britain Locating the radical of a triangular operator algebra BY LAURA MASTRANGELO*, Department of Mathematics, University of Puerto Rico, Mayaguez, PR 00681 PAUL S. MUHLYf Department of Mathematics, University of Iowa, Iowa City, la 52246 AND BARUCH SOLELJ Department of Mathematics, The Technion, 36000 Haifa, Israel (Received 15 March 1993; revised 24 June 1993) Abstract Our primary objective is to give necessary and sufficient conditions for a triangular subalgebra of a groupoid C*-algebra to be semisimple, i.e. to have vanishing Jacobson radical. If, in addition, the subalgebra is the analytic subalgebra determined by a real-valued cocycle on the groupoid, then we can give an explicit description of the radical in terms of the cocycle. As a consequence of this analysis, we are able to determine when certain analytic crossed products are semisimple. 1. Introduction Throughout this note, B will be a separable nuclear C*-algebra containing a diagonal C*-algebra D in the sense of Kumjian[4]. This allows us to write B as C*(G,E) for a suitable second countable, locally compact, Hausdorff, r-discrete, principal groupoid G that is measurewise amenable in the sense of [12] for each quasi- invariant measure on G m . (See [12].) We view G as an equivalence relation on G {0) and we identify G i0) with the diagonal A c G (o) x G {0) . Elements in E will be denoted by lower case Greek letters from the beginning of the alphabet, while elements in G m will be denoted by lower case letters from the end of the Roman alphabet. Elements in G will always be denoted by pairs of elements from G i0) . We identify D with C O (G W ) in the usual way. We also let A denote a not-necessarily self-adjoint, norm-closed subalgebra of B containing D. By the spectral theorem for bimodules [7] there is an open subset P ^ G containing A s,uch that A = A(P), the set of elements in C*(G,E) that are supported on j~ 1 (P), where j:EH> G is the canonical map. The fact that A is an algebra forces P to be a transitive relation, i.e. P satisfies the inclusion PoP £ P. We assume, without loss of generality, that A generates B and therefore that P generates G. Our most definitive results are proved under the assumption that A is triangular in the sense that A (]A* = D, where A* = {a* \aeA}. However, we shall * Supported in part by the National Science Foundation. f Supported in part by grants from the U.S. National Science Foundations and the U.S.-Israel Binational Science Foundation. I Supported by the U.S.-Israel Binational Science Foundation and by the Fund for the Promotion of Research at the Technion.