GNS
Construction for Observable Algebras| GNS
Construction for Observable Algebras |
Gell-Mann–Naimark–Segal Construction in General {Mostly
from an explanation by Abhay, 1985}
* Idea: It allows one to construct, given a star-algebra 
and
a complex valued "positive" linear function f defined on
it, a Hilbert space 
and
a representation of the algebra on the Hilbert space.
$ Construction: If is
a star-algebra, and f : → C a "positive" (i.e.,
non-negative) linear functional,
for all A 
, f(A*A) 
0
(we do not require strict positivity because in many useful physical examples
we
do not have it – f(A) can be the expectation value
of A in
a given state 
,
for example);
- Define I 
as
I:= {A |
f(A*A) = 0}; This is a subspace of ,
in fact it is a left ideal.
- Define V:= /I
(so that the inner product will be positive);
- Then V can be structured as an inner product space defining
{A},{B}
:=
f(A*B), for all A, B
(and, if we complete it in the standard way, it becomes a Hilbert space);
- Also, acts naturally
on V by B({A}):= {BA} (if we get the Hilbert space
by completing V, these operators will in general not be defined
on the
whole ).
* Useful result: If is
a Hilbert space with a cyclic representation of a star-algebra ,
i.e., one such that
0 such
that (for all ,
A such
that = A 0)
,
then this representation is unitarily equivalent to the one obtained by the
GNS construction, with f(A):= 0, A0.
References > s.a. [observable
algebras, operator
theory]; holonomy.
@ General: Reed in(70); Iguri & Castagnino JMP(08)-a0711
[for topological *-algebras].
@ Deformed algebras: Bordemann & Waldmann CMP(98); Waldmann RPMP(01)m.QA/00;
Gozzi
& Reuter IJMPA(94)ht/03.
@ Other generalizations: Hofmann CMP(98);
Naudts & Kuna JPA(01)mp/00 [covariance
systems]; Cariñena et al mp/07/TMP
[alternative].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 apr 2008