- Source: Positive linear functional
In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space
(
V
,
≤
)
{\displaystyle (V,\leq )}
is a linear functional
f
{\displaystyle f}
on
V
{\displaystyle V}
so that for all positive elements
v
∈
V
,
{\displaystyle v\in V,}
that is
v
≥
0
,
{\displaystyle v\geq 0,}
it holds that
f
(
v
)
≥
0.
{\displaystyle f(v)\geq 0.}
In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem.
When
V
{\displaystyle V}
is a complex vector space, it is assumed that for all
v
≥
0
,
{\displaystyle v\geq 0,}
f
(
v
)
{\displaystyle f(v)}
is real. As in the case when
V
{\displaystyle V}
is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace
W
⊆
V
,
{\displaystyle W\subseteq V,}
and the partial order does not extend to all of
V
,
{\displaystyle V,}
in which case the positive elements of
V
{\displaystyle V}
are the positive elements of
W
,
{\displaystyle W,}
by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any
x
∈
V
{\displaystyle x\in V}
equal to
s
∗
s
{\displaystyle s^{\ast }s}
for some
s
∈
V
{\displaystyle s\in V}
to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such
x
.
{\displaystyle x.}
This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.
Sufficient conditions for continuity of all positive linear functionals
There is a comparatively large class of ordered topological vector spaces on which every positive linear form is necessarily continuous.
This includes all topological vector lattices that are sequentially complete.
Theorem Let
X
{\displaystyle X}
be an Ordered topological vector space with positive cone
C
⊆
X
{\displaystyle C\subseteq X}
and let
B
⊆
P
(
X
)
{\displaystyle {\mathcal {B}}\subseteq {\mathcal {P}}(X)}
denote the family of all bounded subsets of
X
.
{\displaystyle X.}
Then each of the following conditions is sufficient to guarantee that every positive linear functional on
X
{\displaystyle X}
is continuous:
C
{\displaystyle C}
has non-empty topological interior (in
X
{\displaystyle X}
).
X
{\displaystyle X}
is complete and metrizable and
X
=
C
−
C
.
{\displaystyle X=C-C.}
X
{\displaystyle X}
is bornological and
C
{\displaystyle C}
is a semi-complete strict
B
{\displaystyle {\mathcal {B}}}
-cone in
X
.
{\displaystyle X.}
X
{\displaystyle X}
is the inductive limit of a family
(
X
α
)
α
∈
A
{\displaystyle \left(X_{\alpha }\right)_{\alpha \in A}}
of ordered Fréchet spaces with respect to a family of positive linear maps where
X
α
=
C
α
−
C
α
{\displaystyle X_{\alpha }=C_{\alpha }-C_{\alpha }}
for all
α
∈
A
,
{\displaystyle \alpha \in A,}
where
C
α
{\displaystyle C_{\alpha }}
is the positive cone of
X
α
.
{\displaystyle X_{\alpha }.}
Continuous positive extensions
The following theorem is due to H. Bauer and independently, to Namioka.
Theorem: Let
X
{\displaystyle X}
be an ordered topological vector space (TVS) with positive cone
C
,
{\displaystyle C,}
let
M
{\displaystyle M}
be a vector subspace of
E
,
{\displaystyle E,}
and let
f
{\displaystyle f}
be a linear form on
M
.
{\displaystyle M.}
Then
f
{\displaystyle f}
has an extension to a continuous positive linear form on
X
{\displaystyle X}
if and only if there exists some convex neighborhood
U
{\displaystyle U}
of
0
{\displaystyle 0}
in
X
{\displaystyle X}
such that
Re
f
{\displaystyle \operatorname {Re} f}
is bounded above on
M
∩
(
U
−
C
)
.
{\displaystyle M\cap (U-C).}
Corollary: Let
X
{\displaystyle X}
be an ordered topological vector space with positive cone
C
,
{\displaystyle C,}
let
M
{\displaystyle M}
be a vector subspace of
E
.
{\displaystyle E.}
If
C
∩
M
{\displaystyle C\cap M}
contains an interior point of
C
{\displaystyle C}
then every continuous positive linear form on
M
{\displaystyle M}
has an extension to a continuous positive linear form on
X
.
{\displaystyle X.}
Corollary: Let
X
{\displaystyle X}
be an ordered vector space with positive cone
C
,
{\displaystyle C,}
let
M
{\displaystyle M}
be a vector subspace of
E
,
{\displaystyle E,}
and let
f
{\displaystyle f}
be a linear form on
M
.
{\displaystyle M.}
Then
f
{\displaystyle f}
has an extension to a positive linear form on
X
{\displaystyle X}
if and only if there exists some convex absorbing subset
W
{\displaystyle W}
in
X
{\displaystyle X}
containing the origin of
X
{\displaystyle X}
such that
Re
f
{\displaystyle \operatorname {Re} f}
is bounded above on
M
∩
(
W
−
C
)
.
{\displaystyle M\cap (W-C).}
Proof: It suffices to endow
X
{\displaystyle X}
with the finest locally convex topology making
W
{\displaystyle W}
into a neighborhood of
0
∈
X
.
{\displaystyle 0\in X.}
Examples
Consider, as an example of
V
,
{\displaystyle V,}
the C*-algebra of complex square matrices with the positive elements being the positive-definite matrices. The trace function defined on this C*-algebra is a positive functional, as the eigenvalues of any positive-definite matrix are positive, and so its trace is positive.
Consider the Riesz space
C
c
(
X
)
{\displaystyle \mathrm {C} _{\mathrm {c} }(X)}
of all continuous complex-valued functions of compact support on a locally compact Hausdorff space
X
.
{\displaystyle X.}
Consider a Borel regular measure
μ
{\displaystyle \mu }
on
X
,
{\displaystyle X,}
and a functional
ψ
{\displaystyle \psi }
defined by
ψ
(
f
)
=
∫
X
f
(
x
)
d
μ
(
x
)
for all
f
∈
C
c
(
X
)
.
{\displaystyle \psi (f)=\int _{X}f(x)d\mu (x)\quad {\text{ for all }}f\in \mathrm {C} _{\mathrm {c} }(X).}
Then, this functional is positive (the integral of any positive function is a positive number). Moreover, any positive functional on this space has this form, as follows from the Riesz–Markov–Kakutani representation theorem.
Positive linear functionals (C*-algebras)
Let
M
{\displaystyle M}
be a C*-algebra (more generally, an operator system in a C*-algebra
A
{\displaystyle A}
) with identity
1.
{\displaystyle 1.}
Let
M
+
{\displaystyle M^{+}}
denote the set of positive elements in
M
.
{\displaystyle M.}
A linear functional
ρ
{\displaystyle \rho }
on
M
{\displaystyle M}
is said to be positive if
ρ
(
a
)
≥
0
,
{\displaystyle \rho (a)\geq 0,}
for all
a
∈
M
+
.
{\displaystyle a\in M^{+}.}
Theorem. A linear functional
ρ
{\displaystyle \rho }
on
M
{\displaystyle M}
is positive if and only if
ρ
{\displaystyle \rho }
is bounded and
‖
ρ
‖
=
ρ
(
1
)
.
{\displaystyle \|\rho \|=\rho (1).}
= Cauchy–Schwarz inequality
=If
ρ
{\displaystyle \rho }
is a positive linear functional on a C*-algebra
A
,
{\displaystyle A,}
then one may define a semidefinite sesquilinear form on
A
{\displaystyle A}
by
⟨
a
,
b
⟩
=
ρ
(
b
∗
a
)
.
{\displaystyle \langle a,b\rangle =\rho (b^{\ast }a).}
Thus from the Cauchy–Schwarz inequality we have
|
ρ
(
b
∗
a
)
|
2
≤
ρ
(
a
∗
a
)
⋅
ρ
(
b
∗
b
)
.
{\displaystyle \left|\rho (b^{\ast }a)\right|^{2}\leq \rho (a^{\ast }a)\cdot \rho (b^{\ast }b).}
Applications to economics
Given a space
C
{\displaystyle C}
, a price system can be viewed as a continuous, positive, linear functional on
C
{\displaystyle C}
.
See also
Positive element – Group with a compatible partial orderPages displaying short descriptions of redirect targets
Positive linear operator – Concept in functional analysis
References
Bibliography
Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Kata Kunci Pencarian:
- Arkea
- Kaidah pendiferensialan
- Titin
- Daftar pemenang Hadiah Ig Nobel
- Positive linear functional
- Linear form
- Continuous linear operator
- State (functional analysis)
- Sublinear function
- Riesz–Markov–Kakutani representation theorem
- Positive linear operator
- Cyclic and separating vector
- List of functional analysis topics
- Cauchy–Schwarz inequality
