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volume 6, issue 5, article 133, 2005.

Received 23 March, 2005;

accepted 23 September, 2005.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

JENSEN’S INEQUALITY FOR CONDITIONAL EXPECTATIONS

FRANK HANSEN

Institute of Economics Copenhagen University Studiestraede 6

1455 Copenhagen K, Denmark.

EMail:Frank.Hansen@econ.ku.dk

c

2000Victoria University ISSN (electronic): 1443-5756 088-05

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Jensen’s Inequality for Conditional Expectations

Frank Hansen

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J. Ineq. Pure and Appl. Math. 6(5) Art. 133, 2005

http://jipam.vu.edu.au

Abstract

We study conditional expectations generated by an abelianC^∗-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.

2000 Mathematics Subject Classification:47A63, 26A51.

Key words: Trace function, Jensen’s inequality, Conditional expectation.

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]

1. Preliminaries

Ann-tuplex= (x₁, . . . , x_n)of elements in aC^∗-algebraAis said to be abelian if the elementsx1, . . . , xnare mutually commuting. We say that an abelian n- tuple xof 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 × · · · ×I_n if the spectrum σ(x_i)of x_i is contained in I_i 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 thatAis realized as operators on a Hilbert space and let

x_i = Z

λ dE_i(λ) i= 1, . . . , n

denote the spectral resolutions of the operators x₁, . . . , x_n. Since the n-tuple x = (x₁, . . . , x_n) is abelian, the spectral measures E₁, . . . , E_n are mutually commuting. We may thus set

E(S₁× · · · ×S_n) =E₁(S₁)· · ·E_n(S_n)

for Borel sets S₁, . . . , S_n inRand extendE to a spectral measure onRⁿ with support inI.Setting

f(x) = Z

f(λ₁, . . . , λ_n)dE(λ₁, . . . , λ_n)

and sincef is continuous, we finally realize thatf(x)is an element inA.

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2. Conditional Expectations

Let C be a separable abelianC^∗-subalgebra of a C^∗-algebraA,and let ϕ be a positive functional onAsuch thatC is contained in the centralizer

A^ϕ ={y∈ A |ϕ(xy) = ϕ(yx) ∀x∈ A}.

The subalgebra is of the formC =C₀(S)for some locally compact metric space S.

Theorem 2.1. There exists a positive linear mapping

(2.1) Φ : M(A)→L^∞(S, µ_ϕ)

on the multiplier algebraM(A)such that

Φ(xy) = Φ(yx) = Φ(x)y, x∈M(A), y ∈ C almost everywhere, and a finite Radon measureµ_ϕonSsuch that

Z

S

z(s)Φ(x)(s)dµ_ϕ(s) =ϕ(zx), z ∈ C, x∈M(A).

Proof. By the Riesz representation theorem there is a finite Radon measureµ_ϕ onSsuch that

ϕ(y) = Z

S

y(s)dµ_ϕ(s), y∈ C =C₀(S).

For each positive elementxin the multiplier algebraM(A)we have 0≤ϕ(yx) = ϕ(y^1/2xy^1/2)≤ kxkϕ(y), y∈ C+.

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The functional y → ϕ(yx) onC consequently defines a Radon measure on S which is dominated by a multiple ofµϕ,and it is therefore given by a unique ele- mentΦ(x)inL^∞(S, µ_ϕ).By linearization this defines a positive linear mapping defined on the multiplier algebra

(2.2) Φ : M(A)→L^∞(S, µ_ϕ)

such that Z

S

z(s)Φ(x)(s)dµ_ϕ(s) =ϕ(zx), z ∈ C, x∈M(A).

Furthermore, since Z

S

z(s)Φ(yx)(s)dµ_ϕ(s) =ϕ(zyx) = Z

S

z(s)y(s)Φ(x)(s)dµ_ϕ(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 A^ϕ and thus ϕ(zxy) = ϕ(yzx),we similarly obtainΦ(xy) = Φ(x)yalmost everywhere.

Note thatΦ(z)(s) = z(s)almost everywhere in S for eachz ∈ C, cf. [6, 4, 5]. With a slight abuse of language we callΦa conditional expectation even though its range is not a subalgebra ofM(A).

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3. Jensen’s inequality

Following the notation in [5] we consider a separableC^∗-algebraAof operators on a (separable) Hilbert spaceH,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 spaceT equipped with a Radon measureν.

We say that the field (a_t)t∈T is weak*-measurable if the functiont → ϕ(a_t)is ν-measurable onT for eachϕ ∈ A^∗;and we say that the field is continuous if the functiont→a_tis continuous [4].

As noted in [5] the field(at)t∈T is weak*-measurable, if and only if for each vectorξ ∈Hthe functiont→a_tξis weakly (equivalently strongly) measurable.

In particular, the composed field(a^∗_tb_t)t∈T is weak*-measurable if both(a_t)t∈T

and(bt)t∈T are weak*-measurable fields.

If for a weak*-measurable field(a_t)t∈T the functiont→ |ϕ(a_t)|is integrable for every stateϕ∈S(A)and the integrals

Z

T

|ϕ(a_t)|dν(t)≤K, ∀ϕ∈S(A)

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

T

a_tdν(t), such that

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ϕ Z

T

a_tdν(t)

= Z

T

ϕ(a_t)dν(t), ∀ϕ ∈ A^∗.

We say in this case that the field(a_t)t∈T is integrable. Finally we say that a field (a_t)t∈T is a unital column field [1,4,5], if it is weak*-measurable and

Z

T

a^∗_ta_tdν(t) = 1.

We note that a C^∗-subalgebra of a separableC^∗-algebra is automatically separable.

Theorem 3.1. Let C be an abelian C^∗-subalgebra of a separable C^∗-algebra A, ϕbe a positive functional on A such thatC is contained in the centralizer A^ϕand let

Φ : M(A)→L^∞(S, µ_ϕ)

be the conditional expectation defined in (2.1). Let furthermoref :I →Rbe a continuous convex function ofn variables defined on a cube, and lett → a_t ∈ M(A)be a unital column field on a locally compact Hausdorff spaceT with a Radon measureν.Ift →x_tis an essentially bounded, weak*-measurable field onT of abeliann-tuples of self-adjoint elements inAin the domain off,then

f(Φ(y₁), . . . ,Φ(y_n))≤Φ Z

T

a^∗_tf(x_t)a_tdν(t) (3.1)

almost everywhere, where then-tupleyinM(A)is defined by setting y= (y₁, . . . , y_n) =

Z

T

a^∗_tx_ta_tdν(t).

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Proof. The subalgebra C is as noted above of the form C = C₀(S) for some locally compact metric spaceS,and since theC^∗-algebraC0(I)is separable we may for almost everysinS define a Radon measureµ_s onI by setting

µs(g) = Z

I

g(λ)dµs(λ) = Φ Z

T

a^∗_tg(x_t)atdµ(t)

(s), g ∈C0(I).

Since

µ_s(1) = Φ Z

T

a^∗_ta_tdµ(t)

= Φ(1) = 1

we observe thatµ_sis a probability measure. If we putg_i(λ) = λ_i then Z

I

g_i(λ)dµ_s(λ) = Φ Z

T

a^∗_tx_ita_tdµ(t)

(s) = Φ(y_i)(s) fori= 1, . . . , nand sincef is convex we obtain

f(Φ(y₁)(s), . . . ,Φ(y_n)(s)) =f Z

I

g₁(λ)dµ_s(λ), . . . , Z

I

g_n(λ)dµ_s(λ)

≤ Z

I

f(g₁(λ), . . . , g_n(λ))dµ_s(λ)

= Z

I

f(λ)dµs(λ)

= Φ Z

T

a^∗_tf(x_t)a_tdµ(t)

(s) for almost allsinS.

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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 → Rbe a continuous convex function ofn variables defined on a cube, and lett →a_t∈B(H)be a unital column field on a locally compact Hausdorff space T with a Radon measure ν.Ift → x_t is a bounded weak*-measurable field onT of abeliann-tuples of self-adjoint operators onH in the domain off,then

f (y1ξ |ξ), . . . ,(ynξ |ξ)

≤ Z

T

a^∗_tf(x_t)atdν(t)ξ|ξ (3.2)

for any unit vectorξ∈H,where then-tupleyis defined by setting y= (y₁, . . . , y_n) =

Z

T

a^∗_tx_ta_tdν(t).

Proof. The statement follows from Theorem3.1by choosingϕas the trace and letting C be the C^∗-algebra generated by the orthogonal projection P on the vector ξ. Then C = C₀(S) whereS = {0,1}, and an element z ∈ C has the representation

z =z(0)P +z(1)(1−P).

The measuredµ_ϕgives unit weight in each of the two points, and the conditional expectationΦis given by

Φ(x)(s) =

( (xξ |ξ) s= 0 Tr(x−P x) s= 1.

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Indeed,

ϕ(zx) = Tr

z(0)P +z(1)(1−P) x

=z(0)Φ(x)(0) +z(1)Φ(x)(0)

= Z

S

z(s)Φ(x)(s)ds

as required. The statement follows by evaluating the functions appearing on each side of the inequality (3.1) at the points= 0.

Remark 1. If we chooseνas a probability measure onT,then the trivial field a_t= 1fort ∈T is unital and (3.2) takes the form

f Z

T

x_1tdν(t)ξ |ξ

, . . . , Z

T

x_ntdν(t)ξ|ξ

≤ Z

T

f(x_t)dν(t)ξ|ξ

for bounded weak*-measurable fields of abeliann-tuplesx_t= (x_1t, . . . , x_nt)of self-adjoint operators in the domain off and unit vectorsξ. By choosingν as an atomic measure with one atom we get a version

f (x₁ξ |ξ), . . . ,(x_nξ|ξ)

≤ f(x)ξ |ξ (3.3)

of the Jensen inequality by Mond and Peˇcari´c [7]. By further considering a direct sum

ξ=

m

M

j=1

ξ_j and x= (x₁, . . . , x_n) =

m

M

j=1

(x_1j, . . . , x_nj)

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we obtain the familiar version

f

m

X

j=1

(x_1jξ_j |ξ_j), . . . ,

m

X

j=1

(x_njξ_j |ξ_j)

!

≤

m

X

j=1

f(x_1j, . . . , x_nj)ξ_j |ξ_j valid for abeliann-tuples(x_1j, . . . , x_nj), j = 1, . . . , mof self-adjoint operators in the domain off and vectorsξ₁, . . . , ξ_m withkξ₁k²+· · ·+kξ_mk² = 1.

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References

[1] H. ARAKI AND F. HANSEN, Jensen’s operator inequality for functions of several variables, Proc. Amer. Math. Soc., 128 (2000), 2075–2084.

[2] L.G. BROWN AND H. KOSAKI, Jensen’s inequality in semi-finite von Neumann algebras, J. Operator Theory, 23 (1990), 3–19.

[3] F. HANSEN, Monotone trace functons of several variables, International Journal of Mathematics, 16 (2005), 777–785.

[4] F. HANSEN AND G.K. PEDERSEN, Jensen’s operator inequality, Bull.

London Math. Soc., 35 (2003), 553–564.

[5] F. HANSEN AND G.K. PEDERSEN, Jensen’s trace inequality in several variables, International Journal of Mathematics, 14 (2003), 667–681.

[6] E. LIEB AND G.K. PEDERSEN, Convex multivariable trace functions, Reviews in Mathematical Physics, 14 (2002), 631–648.

[7] B. MOND AND J.E. PE ˇCARI ´C, On some operator inequalities, Indian Journal of Mathematics, 35 (1993), 221–232.

[8] G.K. PEDERSEN, Analysis Now, Graduate Texts in Mathematics, Vol.

118, Springer Verlag, Heidelberg, 1989, reprinted 1995.

[9] G.K. PEDERSEN, Convex trace functions of several variables on C^∗- algebras, J. Operator Theory, 50 (2003), 157–167.