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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
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(1) |
For a unital  -algebra,
 .
The space of states is a convex set. Let  and  be states, then the convex combination
![$\displaystyle \lambda\Psi_1+(1-\lambda)\Psi_2, \quad \lambda \in [0,1],$](http://images.planetmath.org:8080/cache/objects/4574/l2h/img12.png "$\displaystyle \lambda\Psi_1+(1-\lambda)\Psi_2, \quad \lambda \in [0,1],$") |
(2) |
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.
When a  -algebra is represented on a Hilbert space
 , every unit vector
 determines a (not necessarily pure) state in the form of an expectation value,
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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.
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"state" is owned by mhale.
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Cross-references: functions, distributions, delta functions, expectation value, converse, linear functional, vector, unit vector, Hilbert space, trace, character, equivalent, commutative, extreme points, convex combination, convex set, unital, norm, unit, positive linear functional
There are 9 references to this entry.
This is version 5 of state, born on 2003-08-11, modified 2006-11-21.
Object id is 4574, canonical name is State.
Accessed 11873 times total.
Classification:
| AMS MSC: | 46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras) |
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Pending Errata and Addenda
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