ON RADICALS OF TRIANGULAR OPERATOR-ALGEBRAS

Suppose B is a type I C*-algebra admitting a diagonal D in the sense of Kumjian, and let E be the conditional expectation from B onto D. A subalgebra A of B is called triangular with diagonal D if A boolean AND A* = D. THEOREM: Under the above assumptions the Jacobson radical of A equals the intersection of A with the kernel of the conditional expectation E. Although the statement of the theorem is coordinate free, the proof requires the use of coordinates in essential ways. A theorem by Kumjian allows us to represent every C*-algebra admitting a diagonal as the C*-algebra of a certain groupoid. This enables us to apply the techniques of topological groupoids as developed by Renault and Muhly. A very convenient way of expressing a triangular subalgebra of the C*-algebra of a T-groupoid is given by the Spectral Theorem for Bimodules, due to Qui, which is a descendent of the Spectral Theorem for Bimodules due to Muhly and Solel, and to Muhly, Saito and Solel in the context of von Neumann algebras.