Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 5, Article 133, 2005
JENSEN’S INEQUALITY FOR CONDITIONAL EXPECTATIONS
FRANK HANSEN
I NSTITUTE OF E CONOMICS
C OPENHAGEN U NIVERSITY
S TUDIESTRAEDE 6
1455 C OPENHAGEN K, D ENMARK .
Frank.Hansen@econ.ku.dk
Received 23 March, 2005; accepted 23 September, 2005
Communicated by S.S. Dragomir
A BSTRACT. We study conditional expectations generated by an abelian C ∗ -subalgebra in the
centralizer of a positive functional. We formulate and prove Jensen’s inequality for functions
of several variables with respect to this type of conditional expectations, and we obtain as a
corollary Jensen’s inequality for expectation values.
Key words and phrases: Trace function, Jensen’s inequality, Conditional expectation.
2000 Mathematics Subject Classification. 47A63, 26A51.
1. P RELIMINARIES
An n-tuple x = (x1 , . . . , xn ) of elements in a C ∗ -algebra A is said to be abelian if the
elements x1 , . . . , xn are mutually commuting. We say that an abelian n-tuple x of self-adjoint
elements is in the domain of a real continuous function f of n variables defined on a cube of real
intervals I = I1 × · · · × In if the spectrum σ(xi ) of xi is contained in Ii for each i = 1, . . . , n.
In this situation f (x) is naturally defined as an element in A in the following way. We may
assume that A is realized as operators on a Hilbert space and let
Z
xi = λ dEi (λ)
i = 1, . . . , n
denote the spectral resolutions of the operators x1 , . . . , xn . Since the n-tuple x = (x1 , . . . , xn )
is abelian, the spectral measures E1 , . . . , En are mutually commuting. We may thus set
E(S1 × · · · × Sn ) = E1 (S1 ) · · · En (Sn )
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
This paper is based on the talk given by the author within the “International Conference of Mathematical Inequalities and their Applications,
I”, December 06-08, 2004, Victoria University, Melbourne, Australia [http://rgmia.vu.edu.au/conference].
088-05
2
F RANK H ANSEN
for Borel sets S1 , . . . , Sn in R and extend E to a spectral measure on Rn with support in I.
Setting
Z
f (x) = f (λ1 , . . . , λn ) dE(λ1 , . . . , λn )
and since f is continuous, we finally realize that f (x) is an element in A.
2. C ONDITIONAL E XPECTATIONS
Let C be a separable abelian C ∗ -subalgebra of a C ∗ -algebra A, and let ϕ be a positive functional on A such that C is contained in the centralizer
Aϕ = {y ∈ A | ϕ(xy) = ϕ(yx) ∀x ∈ A}.
The subalgebra is of the form C = C0 (S) for some locally compact metric space S.
Theorem 2.1. There exists a positive linear mapping
Φ : M (A) → L∞ (S, µϕ )
(2.1)
on the multiplier algebra M (A) such that
Φ(xy) = Φ(yx) = Φ(x)y,
x ∈ M (A), y ∈ C
almost everywhere, and a finite Radon measure µϕ on S such that
Z
z(s)Φ(x)(s) dµϕ (s) = ϕ(zx),
z ∈ C, x ∈ M (A).
S
Proof. By the Riesz representation theorem there is a finite Radon measure µϕ on S such that
Z
y(s) dµϕ (s),
y ∈ C = C0 (S).
ϕ(y) =
S
For each positive element x in the multiplier algebra M (A) we have
0 ≤ ϕ(yx) = ϕ(y 1/2 xy 1/2 ) ≤ kxkϕ(y),
y ∈ C+ .
The functional y → ϕ(yx) on C consequently defines a Radon measure on S which is dominated
by a multiple of µϕ , and it is therefore given by a unique element Φ(x) in L∞ (S, µϕ ). By
linearization this defines a positive linear mapping defined on the multiplier algebra
Φ : M (A) → L∞ (S, µϕ )
(2.2)
such that
Z
z ∈ C, x ∈ M (A).
z(s)Φ(x)(s) dµϕ (s) = ϕ(zx),
S
Furthermore, since
Z
Z
z(s)Φ(yx)(s) dµϕ (s) = ϕ(zyx) =
z(s)y(s)Φ(x)(s) dµϕ (s)
S
S
for x ∈ M (A) and z, y ∈ C we derive Φ(yx) = yΦ(x) = Φ(x)y almost everywhere. Since C is
contained in the centralizer Aϕ and thus ϕ(zxy) = ϕ(yzx), we similarly obtain Φ(xy) = Φ(x)y
almost everywhere.
Note that Φ(z)(s) = z(s) almost everywhere in S for each z ∈ C, cf. [6, 4, 5]. With a slight
abuse of language we call Φ a conditional expectation even though its range is not a subalgebra
of M (A).
J. Inequal. Pure and Appl. Math., 6(5) Art. 133, 2005
http://jipam.vu.edu.au/
J ENSEN ’ S I NEQUALITY
FOR
C ONDITIONAL E XPECTATIONS
3
3. J ENSEN ’ S INEQUALITY
Following the notation in [5] we consider a separable C ∗ -algebra A of operators on a (separable) Hilbert space H, and a field (at )t∈T of operators in the multiplier algebra
M (A) = {a ∈ B(H) | aA + Aa ⊆ A}
defined on a locally compact metric space T equipped with a Radon measure ν. We say that
the field (at )t∈T is weak*-measurable if the function t → ϕ(at ) is ν-measurable on T for each
ϕ ∈ A∗ ; and we say that the field is continuous if the function t → at is continuous [4].
As noted in [5] the field (at )t∈T is weak*-measurable, if and only if for each vector ξ ∈ H
the function t → at ξ is weakly (equivalently strongly) measurable. In particular, the composed
field (a∗t bt )t∈T is weak*-measurable if both (at )t∈T and (bt )t∈T are weak*-measurable fields.
If for a weak*-measurable field (at )t∈T the function t → |ϕ(at )| is integrable for every state
ϕ ∈ S(A) and the integrals
Z
|ϕ(at )| dν(t) ≤ K,
∀ϕ ∈ S(A)
T
are uniformly bounded by some constant K, then there is a unique element (a C ∗ -integral in
Pedersen’s terminology [8, 2.5.15]) in the multiplier algebra M (A), designated by
Z
at dν(t),
T
such that
Z
ϕ
Z
at dν(t) =
ϕ(at ) dν(t),
T
∀ϕ ∈ A∗ .
T
We say in this case that the field (at )t∈T is integrable. Finally we say that a field (at )t∈T is a
unital column field [1, 4, 5], if it is weak*-measurable and
Z
a∗t at dν(t) = 1.
T
We note that a C ∗ -subalgebra of a separable C ∗ -algebra is automatically separable.
Theorem 3.1. Let C be an abelian C ∗ -subalgebra of a separable C ∗ -algebra A, ϕ be a positive
functional on A such that C is contained in the centralizer Aϕ and let
Φ : M (A) → L∞ (S, µϕ )
be the conditional expectation defined in (2.1). Let furthermore f : I → R be a continuous
convex function of n variables defined on a cube, and let t → at ∈ M (A) be a unital column
field on a locally compact Hausdorff space T with a Radon measure ν. If t → xt is an essentially
bounded, weak*-measurable field on T of abelian n-tuples of self-adjoint elements in A in the
domain of f, then
Z
∗
(3.1)
f (Φ(y1 ), . . . , Φ(yn )) ≤ Φ
at f (xt )at dν(t)
T
almost everywhere, where the n-tuple y in M (A) is defined by setting
Z
y = (y1 , . . . , yn ) =
a∗t xt at dν(t).
T
Proof. The subalgebra C is as noted above of the form C = C0 (S) for some locally compact
metric space S, and since the C ∗ -algebra C0 (I) is separable we may for almost every s in S
define a Radon measure µs on I by setting
Z
Z
∗
µs (g) = g(λ) dµs (λ) = Φ
at g(xt )at dµ(t) (s),
g ∈ C0 (I).
I
J. Inequal. Pure and Appl. Math., 6(5) Art. 133, 2005
T
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F RANK H ANSEN
Since
Z
a∗t at
µs (1) = Φ
dµ(t) = Φ(1) = 1
T
we observe that µs is a probability measure. If we put gi (λ) = λi then
Z
Z
∗
gi (λ) dµs (λ) = Φ
at xit at dµ(t) (s) = Φ(yi )(s)
I
T
for i = 1, . . . , n and since f is convex we obtain
Z
Z
f (Φ(y1 )(s), . . . , Φ(yn )(s)) = f
g1 (λ) dµs (λ), . . . , gn (λ) dµs (λ)
I
I
Z
≤ f (g1 (λ), . . . , gn (λ)) dµs (λ)
I
Z
= f (λ) dµs (λ)
I
Z
∗
at f (xt )at dµ(t) (s)
=Φ
T
for almost all s in S.
The following corollary is known as “Jensen’s inequality for expectation values”. It was
formulated (for continuous fields) in the reference [3], where a more direct proof is given.
Corollary 3.2. Let f : I → R be a continuous convex function of n variables defined on a cube,
and let t → at ∈ B(H) be a unital column field on a locally compact Hausdorff space T with
a Radon measure ν. If t → xt is a bounded weak*-measurable field on T of abelian n-tuples of
self-adjoint operators on H in the domain of f, then
Z
∗
(3.2)
at f (xt )at dν(t)ξ | ξ
f (y1 ξ | ξ), . . . , (yn ξ | ξ) ≤
T
for any unit vector ξ ∈ H, where the n-tuple y is defined by setting
Z
y = (y1 , . . . , yn ) =
a∗t xt at dν(t).
T
Proof. The statement follows from Theorem 3.1 by choosing ϕ as the trace and letting C be the
C ∗ -algebra generated by the orthogonal projection P on the vector ξ. Then C = C0 (S) where
S = {0, 1}, and an element z ∈ C has the representation
z = z(0)P + z(1)(1 − P ).
The measure dµϕ gives unit weight in each of the two points, and the conditional expectation Φ
is given by
(
(xξ | ξ)
s=0
Φ(x)(s) =
Tr(x − P x) s = 1.
Indeed,
ϕ(zx) = Tr
z(0)P + z(1)(1 − P ) x
= z(0)Φ(x)(0) + z(1)Φ(x)(0)
Z
=
z(s)Φ(x)(s) ds
S
J. Inequal. Pure and Appl. Math., 6(5) Art. 133, 2005
http://jipam.vu.edu.au/
J ENSEN ’ S I NEQUALITY
FOR
C ONDITIONAL E XPECTATIONS
5
as required. The statement follows by evaluating the functions appearing on each side of the
inequality (3.1) at the point s = 0.
Remark 3.3. If we choose ν as a probability measure on T, then the trivial field at = 1 for
t ∈ T is unital and (3.2) takes the form
Z
Z
Z
f
x1t dν(t)ξ | ξ , . . . ,
xnt dν(t)ξ | ξ
≤
f (xt ) dν(t)ξ | ξ
T
T
T
for bounded weak*-measurable fields of abelian n-tuples xt = (x1t , . . . , xnt ) of self-adjoint
operators in the domain of f and unit vectors ξ. By choosing ν as an atomic measure with one
atom we get a version
(3.3)
f (x1 ξ | ξ) , . . . , (xn ξ | ξ) ≤ f (x)ξ | ξ
of the Jensen inequality by Mond and Pečarić [7]. By further considering a direct sum
m
m
M
M
ξ=
ξj and x = (x1 , . . . , xn ) =
(x1j , . . . , xnj )
j=1
j=1
we obtain the familiar version
!
m
m
m X
X
X
f
(x1j ξj | ξj ), . . . ,
(xnj ξj | ξj ) ≤
f (x1j , . . . , xnj )ξj | ξj
j=1
j=1
j=1
valid for abelian n-tuples (x1j , . . . , xnj ), j = 1, . . . , m of self-adjoint operators in the domain
of f and vectors ξ1 , . . . , ξm with kξ1 k2 + · · · + kξm k2 = 1.
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J. Inequal. Pure and Appl. Math., 6(5) Art. 133, 2005
http://jipam.vu.edu.au/