A state on a -algebra is a positive linear functional , for all , with unit norm. The norm of a positive linear functional is defined by
For a unital -algebra, .
The space of states is a convex set. Let and be states, then the convex combination
is also a state.
A state is pure if it is not a convex combination of two other states. Pure states are the extreme points of the convex set of states. A pure state on a commutative -algebra is equivalent to a character.
A state is called a tracial state if it is also a trace.
In physics, it is common to refer to such states by their vector rather than the linear functional . The converse is not always true; not every state need be given by an expectation value. For example, delta functions (which are distributions not functions) give pure states on , but they do not correspond to any vector in a Hilbert space (such a vector would not be square-integrable).
- 1 G. Murphy, -Algebras and Operator Theory. Academic Press, 1990.
|Date of creation||2013-03-22 13:50:18|
|Last modified on||2013-03-22 13:50:18|
|Last modified by||mhale (572)|