(EC Lance obtained this result independently [15].) This distance formula h proven to be very useful in investigating problems involving compact perturba and similarity theory for nests [1, 2, 4, 11, 12, 15, 17]. Certain other reflexive bras have been shown to possess the property that there exists a constant К s that d (T, si)^ ATsup {||/>-L TT^ d: P e lat si} for all T in L (H)[7, 8, 9, 10, 13]. Simple examples show that this constant need not be 1. K. Davidson asked in [10] whether every reflexive algebra has such a dis tance estimate for some constant K.(This question was also posed in [13, 15, 16].) In [5], W. Arveson conjectured that the answer to this question is no. In this paper we verify this conjecture by producing an explicit example of a reflexive algebra that does not satisfy a distance estimate for any K. We also give examples of points of discontinuity of the maps if-» alg if and si~> lat si, where the topology for both lattices and algebras is that induced by the Hausdorff metric. We would like to thank W. Arveson for several helpful conversations con cerning this paper, and in particular for pointing out to us that if У is a reflexive subspace of L (H), then is a reflexive subalgebra of L (H©//). We use this observation to construct reflexive algebras with certain properties by first constructing reflexive subspaces with ana logous properties. The research for this paper was done while both authors were visiting the University of California at Berkeley.