Extreme points in triangular UHF algebras

Abstract. We examine the strongly extreme point structure of the unit balls of triangular UHF algebras. The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal. In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points. Results concerning the geometry of the unit ball have a long history both in Banach space theory and in the theory of operator algebras. This geometry can be aliated with structural and algebraic properties. Moreover, dierences in geo-metric properties can prove useful in classication problems. Two fundamental results are the Russo-Dye Theorem and Kadison’s Theorem on isometries. More recently, there has been interest in the unit balls of nonselfadjoint operator algebras, especially nest algebras [1, 2, 3, 4, 15, 16]. This paper concerns the unit balls of triangular UHF algebras. These and th