A non-selfadjoint Russo-Dye Theorem

One of the well-known results in the theory of C* algebras is the Russo-Dye Theorem [19]: given a C* algebra .~r the closed convex hull of the unitary elements in ~r equals the closed unit ball of ~r This result was later refined by Gardner and reached its final form by Kadison and Pedersen; today it is known that every operator in a C* algebra ~r whose norm is less than 1, is the average of unitaries from A. The Russo-Dye Theorem initiated the theory of unitary rank in selfadjoint operator algebras. If ~r is an operator algebra, the unitary rank of an element A E ~r is defined as the smallest number for which there is a convex combination of unitaries from ~r of length u(A) and equaling A. If no such decomposition exists (in particular if liAII > 1) we define u(A) = oo. The literature on unitary rank is vast. The earliest result is due to Murray and yon Neumann who proved that any selfadjoint operator of norm I or less is the mean of two unitary operators ([12] p. 239, 1937). The first systematic study was given by R. Kadison and G. Pedersen [8] in 1984 (previous work in the field included contributions by Popa [15], Robertson [17], Gardner [6] and others). In 1986, C. Olsen and G. Pedersen [14] characterized all elements in a factor von Neumarm algebra with finite unitary rank. In the general case of a C*-algebra, a characterization was obtained by Rordam in his important paper [18]. For more details and further information on the theory of unitary rank we refer to the excellent articles of U. Haagerup [7] and M. Rordam [ 18]. In the first section of the present paper, we prove a Russo-Dye type Theorem for infinite multiplicity nest algebras. The techniques employed in the proof of our result are different from that of Gardner and Kadison-Pedersen. To our knowledge, this is the first result of this type, for non-selfadjoint operator algebras and clearly initiates the unitary rank theory for such algebras.