REGULARITY FOR OPERATOR-ALGEBRAS ON A HILBERT-SPACE

Four notions of regularity for operator algebras are introduced. An al gebra A is called 1-regular if for any two linearly independent vector s x, y is-an-element-of H there is an a is-an-element-of A such that a x = 0 and ay not-equal 0. We show that the only weakly closed transiti ve 1-regular algebra is B(H) thus providing a natural generalization o f the Rickart-Yood density theorem. We construct an example of a 1-reg ular algebra which contains no nonzero compact operators. This example is related to the ''thin set'' phenomena of classical harmonic analys is.