We study conditional expectations generated by an abelianC∗-subalgebra in the centralizer of a positive functional

http://jipam.vu.edu.au/

Volume 6, Issue 5, Article 133, 2005

JENSEN’S INEQUALITY FOR CONDITIONAL EXPECTATIONS

FRANK HANSEN INSTITUTE OFECONOMICS

COPENHAGENUNIVERSITY

STUDIESTRAEDE6

1455 COPENHAGENK, DENMARK. Frank.Hansen@econ.ku.dk

Received 23 March, 2005; accepted 23 September, 2005 Communicated by S.S. Dragomir

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.

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

2000 Mathematics Subject Classification. 47A63, 26A51.

1. PRELIMINARIES

An n-tuple x = (x₁, . . . , x_n) of elements in a C^∗-algebra A is said to be abelian if the elementsx₁, . . . , x_n are mutually commuting. We say that an abeliann-tuplexof self-adjoint elements is in the domain of a real continuous functionfofnvariables defined on a cube of real intervalsI =I1× · · · ×Inif the spectrumσ(xi)ofxiis contained inIifor 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 operatorsx1, . . . , xn.Since then-tuplex = (x1, . . . , xn) is abelian, the spectral measuresE₁, . . . , E_nare mutually commuting. We may thus set

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

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

for Borel sets S₁, . . . , S_n in R and extend E to a spectral measure on Rⁿ with support in I.

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.

2. CONDITIONAL EXPECTATIONS

LetC be a separable abelianC^∗-subalgebra of aC^∗-algebraA,and letϕ be a positive func- tional 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 spaceS.

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_ϕ onS such 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 measureµϕonSsuch 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₊.

The functionaly→ϕ(yx)onCconsequently defines a Radon measure onSwhich is dominated by a multiple of µ_ϕ, and it is therefore given by a unique element Φ(x) in L^∞(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)

forx∈M(A)andz, y ∈ C we deriveΦ(yx) = yΦ(x) = Φ(x)yalmost everywhere. SinceC is contained in the centralizerA^ϕand thusϕ(zxy) =ϕ(yzx),we similarly obtainΦ(xy) = Φ(x)y

almost everywhere.

Note thatΦ(z)(s) =z(s)almost everywhere inSfor 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).

3. JENSEN’S INEQUALITY

Following the notation in [5] we consider a separableC^∗-algebraAof operators on a (sepa- rable) 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 measure ν. We say that the field(a_t)t∈T is weak*-measurable if the functiont → ϕ(a_t)isν-measurable onT for each ϕ∈ A^∗;and we say that the field is continuous if the functiont →atis continuous [4].

As noted in [5] the field(at)t∈T is weak*-measurable, if and only if for each vectorξ ∈ H the 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(b_t)_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 state ϕ∈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 algebraM(A),designated by

Z

T

a_tdν(t), such that

ϕ Z

T

atdν(t)

= Z

T

ϕ(at)dν(t), ∀ϕ ∈ A^∗.

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

T

a^∗_ta_tdν(t) = 1.

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

Theorem 3.1. LetCbe an abelianC^∗-subalgebra of a separableC^∗-algebraA, ϕbe a positive functional onAsuch thatC is contained in the centralizerA^ϕand let

Φ : M(A)→L^∞(S, µ_ϕ)

be the conditional expectation defined in (2.1). Let furthermore f : I → R be a continuous convex function ofnvariables defined on a cube, and let t → 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 inA in 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).

Proof. The subalgebra C is as noted above of the formC = C₀(S) for some locally compact metric space S, and since the C^∗-algebraC₀(I) is separable we may for almost every s inS define a Radon measureµ_sonI by setting

µ_s(g) = Z

I

g(λ)dµ_s(λ) = Φ Z

T

a^∗_tg(x_t)a_tdµ(t)

(s), g ∈C₀(I).

Since

µ_s(1) = Φ Z

T

a^∗_ta_tdµ(t)

= Φ(1) = 1 we observe thatµsis a probability measure. If we putgi(λ) = λithen

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(yn)(s)) =f Z

I

g1(λ)dµs(λ), . . . , Z

I

gn(λ)dµs(λ)

≤ Z

I

f(g1(λ), . . . , gn(λ))dµs(λ)

= Z

I

f(λ)dµs(λ)

= Φ Z

T

a^∗_tf(x_t)a_tdµ(t)

(s)

for almost allsinS.

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. Letf :I →Rbe a continuous convex function ofnvariables defined on a cube, and lett →a_t ∈B(H)be a unital column field on a locally compact Hausdorff spaceT with a Radon measureν.Ift →x_tis a bounded weak*-measurable field onT of abeliann-tuples of self-adjoint operators onH in the domain off,then

f (y₁ξ|ξ), . . . ,(y_nξ|ξ)

≤ Z

T

a^∗_tf(x_t)a_tdν(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 Theorem 3.1 by choosingϕas the trace and lettingC be the C^∗-algebra generated by the orthogonal projectionP on the vectorξ. ThenC = C0(S)where S ={0,1},and an elementz ∈ 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 expectationΦ is given by

Φ(x)(s) =

( (xξ |ξ) s= 0 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

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 3.3. If we choose ν as a probability measure onT, then the trivial fieldat = 1for t∈T is unital and (3.2) takes the form

f Z

T

x1tdν(t)ξ|ξ

, . . . , Z

T

xntdν(t)ξ |ξ

≤ Z

T

f(x_t)dν(t)ξ|ξ

for bounded weak*-measurable fields of abelian n-tuples x_t = (x_1t, . . . , x_nt) of self-adjoint operators in the domain off and unit vectorsξ.By choosing νas 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) 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 abelian n-tuples(x1j, . . . , xnj), j = 1, . . . , mof self-adjoint operators in the domain off and vectorsξ₁, . . . , ξ_mwithkξ₁k²+· · ·+kξ_mk² = 1.

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