# IRREDUCIBLE TRIANGULAR ALGEBRAS

#### Abstract

The thesis is devoted to the study of the structure of irreducible triangular algebras generated by a maximal abelian algebra and an ordered semigroup G of unitary operators acting on .^ The theory of triangular operator algebras was initiated by Kadison^and Singer (1960). With H a complex Hilbert space and B(H) the algebra^of all bounded operators on it, a subalgebra of B(H) such that (INTERSECT) *^is maximal abelian in B(H) is said to be triangular. It is said to be maximal triangular if it is maximal with this property.^ If H is finite dimensional and is maximal triangular, an (ordered) orthonormal basis can be chosen so that appears as the algebra of (upper) triangular matrices.^ A projection E is said to be a hull (in ) when E is invariant under^each operator in . The von Neumann algebra generated by the hulls^is called the core. The triangular algebras for which the core coincides^with (INTERSECT) * are called hyperreducible. These algebras were studied in detail by Kadison and Singer.^ They also introduced the irreducible triangular algebras--those triangular algebras whose core consists of scalar multiples of the identity operator I--and proved that the algebra generated by a maximal abelian algebra and a unitary operator U acting ergodically on , is irreducible triangular.^ We extend this construction to include free and ergodic actions of^ordered semigroups G of unitary operators on a maximal abelian^algebra .^ The construction provides us with the class of irreducible triangular algebras whose detailed analysis is the subject of the thesis.^ Our techniques allow us to decide when the acting semigroup is singly generated. This is done by relating the structure of the algebra to properties of the ordering on the semigroup G.^ Among other techniques used in this study is the Arveson expectation and an operator-algebra version of Dye's F-analysis. Combining this analysis with estimates (Theorm 2.14 and Lemma 2.15) we are able to detect unitary operators in the algebra generated by the ordered semigroup and the maximal abelian algebra on which it acts (as distinguished from its norm closure--cf. Theorem 2.16 and Corollary 2.17).^ We define a homomorphism from the irreducible triangular algebra into B(K), for some Hilbert space K, to be a linear multiplicative map whose restriction to is self adjoint.^ In Chapter III we study these homomorphisms and obtain results that enable us to construct examples of non isomorphic irreducible triangular algebras.^ We find conditions under which the homomorphism extends to a *-homomorphism on the self adjoint algebra generated by and, further, to a *-homomorphism on the C*-algebra generated by .^ Using this, and the theory of spectral spaces of a group of *-automorphisms on a C*-algebra (introduced by Arveson in 1974), we analyse the group of *-automorphisms of the C*-algebra , generated by , that leave the operators in (the maximal abelian algebra) fixed.^ We study the correspondence between these *-automorphisms and the skew adjoint derivations on that vanish on and are able to decide when such a derivation is bounded. ^

#### Subject Area

Mathematics

#### Recommended Citation

SOLEL, BARUCH, "IRREDUCIBLE TRIANGULAR ALGEBRAS" (1981). *Dissertations available from ProQuest*. AAI8117855.

https://repository.upenn.edu/dissertations/AAI8117855