## Mathematics, Department of

#### Date of this Version

April 2008

#### Abstract

The generalized state space of a commutative *C**-algebra, denoted *S _{H}*(

*C*(

*X*)), is the set of positive unital maps from

*C*(

*X*) to the algebra

*B*(

*H*) of bounded linear operators on a Hilbert space

*H*.

*C**-convexity is one of several non-commutative analogs of convexity which have been discussed in this context. We show that a

*C**-extreme point of

*S*(

_{H}*C*(

*X*)) satisfies a certain spectral condition on the operators in the range of an associated measure, which is a positive operator-valued measure on

*X*. We then show that

*C**-extreme maps from

*C*(

*X*) into

*K*

^{+}, the

*C**-algebra generated by the compact and scalar operators, are multiplicative, generalizing a result of D. Farenick and P. Morenz.

Adviser: Professor David R. Pitts

## Comments

A DISSERTATION Presented to the Faculty of the Graduate College at the University of Nebraska in Partial Fulfillment of Requirements for the Degree of Doctor of Philosophy, Major: Mathematics. Under the Supervision of Professor David R. Pitts.

Lincoln, Nebraska: May, 2008

Copyright © 2008 Martha Gregg.