Axiomatic Symmetric Cones and the Thompson metricconstantin.vernicos.org/Articles/Hilbgeom.pdfThe Order For a normal cone Ω, the relation x ≤ y if and only if y −x ∈ Ω is a

Axiomatic Symmetric Cones and theThompson metric

Jimmie Lawson

Work with Yongdo Lim

Louisiana State University

Axiomatic Symmetric Cones and the Thompson metric – p. 1/25

Background

Let A be a unital C∗-algebra with identity e, and let A+ be

the set of positive invertible elements of A. It is known thatA

+ is an open convex cone in the space H(A) of hermitianelements. The geometry of A

+ has been studied by severalauthors. One particular focus in this geometry has been thestudy of appropriate non-positive curvature properties.


Background

Let A be a unital C∗-algebra with identity e, and let A+ be

the set of positive invertible elements of A. It is known thatA

+ is an open convex cone in the space H(A) of hermitianelements. The geometry of A

+ has been studied by severalauthors. One particular focus in this geometry has been thestudy of appropriate non-positive curvature properties.

One approach has been to endow A+ with a natural Finsler

structure and metric and use this for a substitute for theRiemannian geometry commonly considered infinite-dimensional examples. For example,Andruchow-Corach-Stojanoff and Corach-Porta-Recht haveshown the convexity of the distance function along twodistinct geodesics and its equivalence to the well-knownLoewner-Heinz inequality.


Background II

K.-H. Neeb established an appropriate differentialgeometric notion of seminegative (equal non-positive)curvature for certain classes of Finsler manifolds,particularly the infinite-dimensional symmetric spaces,including the case of A

+.


Background II

K.-H. Neeb established an appropriate differentialgeometric notion of seminegative (equal non-positive)curvature for certain classes of Finsler manifolds,particularly the infinite-dimensional symmetric spaces,including the case of A

+.

Our approach has been somewhat different from either ofthe preceding. We replace the differential geometricstructure by the structure of a metric symmetric spaceendowed with a midpoint operation and studyseminegativity via the metric. We have obtained theconvexity of the metric for such symmetric spaces withweaker metric assumptions than those enjoyed by theFinsler metric on A

+.


Overview

The main goal of this presentation is to present a verygeneral framework, that of a general notion of a symmetriccone, in which one can derive and study variousinequalities, such as those familiar from operator theory.


Overview

The main goal of this presentation is to present a verygeneral framework, that of a general notion of a symmetriccone, in which one can derive and study variousinequalities, such as those familiar from operator theory.

Our notion of a symmetric cone is that of an open normalcone in a Banach space equipped with an axiomaticsymmetric structure that is appropriately connected to theconal structure. All of this of course, needs to be madeprecise.


Proper and Normal Cones

Let V be a Banach space and let Ω henceforth denote anon-empty proper open convex cone of V : tΩ ⊂ Ω for allt > 0, Ω + Ω ⊂ Ω, and Ω ∩ −Ω = 0, where Ω denotes theclosure of Ω.


Proper and Normal Cones

Let V be a Banach space and let Ω henceforth denote anon-empty proper open convex cone of V : tΩ ⊂ Ω for allt > 0, Ω + Ω ⊂ Ω, and Ω ∩ −Ω = 0, where Ω denotes theclosure of Ω.

We further assume that Ω is a normal cone: there exists aconstant K with ||x|| ≤ K||y|| for all x, y ∈ Ω with x ≤ y.


The Order

For a normal cone Ω, the relation

x ≤ y if and only if y − x ∈ Ω

is a partial order. We write x < y if y − x ∈ Ω.


The Order

For a normal cone Ω, the relation

x ≤ y if and only if y − x ∈ Ω

is a partial order. We write x < y if y − x ∈ Ω.

Any member ε of Ω is an order unit for the ordered space(V,≤), and the cone is normal if and only if the order unitnorm determined by ε

‖x‖ε = infM > 0 : −Mε ≤ x ≤ Mε

is compatible, i.e., determines the topology of V . In thiscase 0 ≤ x ≤ y implies ‖x‖ε ≤ ‖y‖ε, that is, we may (and do)assume without loss of generality that K = 1.


The Thompson Metric

A. C. Thompson has shown that Ω is a complete metricspace with respect to the Thompson (part) metric

d(x, y) = maxlog M(x/y), log M(y/x),

where M(x/y) := infλ > 0 : x ≤ λy.


The Thompson Metric

A. C. Thompson has shown that Ω is a complete metricspace with respect to the Thompson (part) metric

d(x, y) = maxlog M(x/y), log M(y/x),

where M(x/y) := infλ > 0 : x ≤ λy.The Thompson metric can be alternatively realized as anappropriately defined Finsler length metric (Nussbaum): forx ∈ Ω and v ∈ V = TxΩ, |v|x(= inft > 0 : −tx ≤ v ≤ tx).The Thompson part metric d(x, y) agrees with the Finslerdistance from x to y:

d(x, y) = inf∫ 1

0|γ′(t)|γ(t) : γ ∈ S, γ(0) = x, γ(1) = y,

where S =the set of piecewise C1-maps γ : [0, 1] → Ω.Axiomatic Symmetric Cones and the Thompson metric – p. 7/25

Strict Symmetric Sets

A strict symmetric set is a triple (X, •,#) with left translationSxy := x • y representing the point symmetry through x,satisfying for all a, b, c ∈ X:

(S1) a • a = a (Saa = a);

(S2) a • (a • b) = b (SaSa = idX);

(S3) a • (b • c) = (a • b) • (a • c) (SaSb = SSabSa);

(S4) (strictness) the equation x • a = b (Sxa = b) has aunique solution x = a#b called the midpoint ofsymmetry or mean of a and b.


Strict Symmetric Sets

A strict symmetric set is a triple (X, •,#) with left translationSxy := x • y representing the point symmetry through x,satisfying for all a, b, c ∈ X:

(S1) a • a = a (Saa = a);

(S2) a • (a • b) = b (SaSa = idX);

(S3) a • (b • c) = (a • b) • (a • c) (SaSb = SSabSa);

(S4) (strictness) the equation x • a = b (Sxa = b) has aunique solution x = a#b called the midpoint ofsymmetry or mean of a and b.

The binary operations • and # uniquely determine eachother via

c = a#b ⇔ c • a = b.


The Loos Axioms

The Loos axioms for a symmetric space posit a topological(smooth) space X equipped with a continuous (smooth)binary operation • satisfying (S1)-(S3) and(S4’) For each x ∈ X, the fixed points of Sx are isolated.

Our variant (S4) implies (S4’) since under (S4) the onlyfixed point of Sx is x itself. Thus the axioms for a strictsymmetric space have a purely algebraic formulation.


Twisted Subgroups

A twisted subgroup P of a group G is a subset containingthe identity e, closed under inversion, and under(a, b) 7→ aba. With respect to a • b := ab−1a (sometimescalled the “core" operation), P satisfies (S1), (S2), and (S3).


Twisted Subgroups


The twisted subgroup P is a strict symmetric set if and onlyif it is uniquely 2-divisible (every element of P has a uniquesquare root in P ). In this casea#b = a1/2(a−1/2ba−1/2)1/2a1/2.


Twisted Subgroups


The twisted subgroup P is a strict symmetric set if and onlyif it is uniquely 2-divisible (every element of P has a uniquesquare root in P ). In this casea#b = a1/2(a−1/2ba−1/2)1/2a1/2.

Example. For a C∗-algebra A, the positive partA

+ = xx∗ : x ∈ A ∩ G(A) is such a twisted subgroup ofthe group G(A) of invertible elements.


A Converse

Let ε be a distinguished point of a strict symmetric set(P, •,#). For x ∈ P , define the displacement Q(x) : P → Pby Q(x) = SxSε. The displacements are isomorphisms of Pand generate the displacement group G(P ).


A Converse

Let ε be a distinguished point of a strict symmetric set(P, •,#). For x ∈ P , define the displacement Q(x) : P → Pby Q(x) = SxSε. The displacements are isomorphisms of Pand generate the displacement group G(P ).

The map x 7→ Q(x) from P to G(P ), called the quadraticrepresentation, carries P to Q(P ), a uniquely 2-divisibletwisted subgroup of G(P ), and is an isomorphism onto Q(P )endowed with the core operation and corresponding meanoperation.


Weighted Means

The core of the group (R,+) has operations

λ • µ = λ − µ + λ = 2λ − µ; λ#µ =λ + µ

2,

and these operations restrict to a strict symmetric structureon the dyadic rationals D.


Weighted Means

The core of the group (R,+) has operations

λ • µ = λ − µ + λ = 2λ − µ; λ#µ =λ + µ

2,

and these operations restrict to a strict symmetric structureon the dyadic rationals D.

Given x 6= y in a strict symmetric set P , there exists aunique morphism of (D, •,#) to P taking 0 to x and 1 to y.We denote the image of t ∈ D by x#ty, called thet-weighted mean of x and y. Note x#1/2y = x#y.


Powers

Dyadic powers for (P, •,#, ε) arise as a special case ofweighted means: xt = ε#tx for x ∈ P , t ∈ D.In the case of a uniquely 2-divisible twisted subgroupequipped with the core operation and with ε the identity, thisnotion agrees with powers in the group. Thus powers maybe alternatively defined through the quadraticrepresentation.


Powers

Dyadic powers for (P, •,#, ε) arise as a special case ofweighted means: xt = ε#tx for x ∈ P , t ∈ D.In the case of a uniquely 2-divisible twisted subgroupequipped with the core operation and with ε the identity, thisnotion agrees with powers in the group. Thus powers maybe alternatively defined through the quadraticrepresentation.

For s, t ∈ D, x ∈ P , exponents satisfy the laws

xst = xst, xs • xt = x2s−t, xs#xt = x(s+t)/2.


Symmetric Spaces

A pointed strict symmetric space with convex metric is apointed strict symmetric set P equipped with a completemetric d(·, ·) satisfying for all x, y ∈ P and g ∈ G(P )

(i) d(g.x, g.y) = d(x, y),

(ii) d(x−1, y−1) = d(x, y),

(iii) d(ε, x1/2) = (1/2)d(ε, x),

(iv) d(x1/2, y1/2) ≤ 12d(x, y) (Busemann nonpositive

curvature),

(v) x 7→ x2 : P → P is continuous.

A strict symmetric space with convex metric is a strictsymmetric set equipped with a complete metric that is apointed strict symmetric space with convex metric withrespect to some pointing.


Symmetric Space Properties

Let (P, •,#) be a pointed strict symmetric space withconvex metric.

1. For x 6= y, ∃! αx,y : (R, •,#) → (P, •,#) a continuoushomomorphism that is also a metric geodesic such thatαx,y(0) = x and αx,y(1) = y.

2. For every pair (β, γ) of such geodesics, the real function

t 7→ d(β(t), γ(t))

is a convex function (Busemann NPC II).

3. The maps (x, y) 7→ x • y : P × P → P and(t, x, y) 7→ αx,y(t) := x#ty : R × P × P → P arecontinuous. The element x#ty is called the t-weightedmean of x and y. Note that x#y = x#1/2y.


Cones With Symmetric Structures

Key Lemma. Let Ω be an open normal cone in a Banachspace V. Suppose in addition that (Ω, •,#, ε) is a pointedstrict symmetric set such that the displacementsQ(x) : Ω → Ω are positively homogeneous for all x ∈ Ω.Then (λx)−1 = 1

λx−1 and λtε = (λε)t for all λ > 0 and alldyadic rationals t. Moreover µx#tλx = µ1−tλtx for all x ∈ Ω,µ, λ > 0 and all dyadic rationals t. Furthermore, thefollowing conditions are equivalent:

(i) (λx)1/2 =√

λx1/2 for all x ∈ Ω and λ > 0;

(ii) (λx)t = λtxt for any dyadic rational t and x ∈ Ω andλ > 0;

(iii) Q(λx) = λ2Q(x) for any x ∈ Ω and λ > 0.


Key Theorem I

Theorem. Let Ω be an open normal cone in a Banachspace V. Suppose that there is a pointed strict symmetricstructure on Ω satisfying

(i) ε#x = x1/2 ≤ (ε + x)/2.

(ii) the squaring map x 7→ x2 = Q(x)ε is continuous (in therelative norm topology of Ω).

(iii) every basic displacement Q(x) is continuous and linear(that is, additive and positively homogeneous) on Ω.

Then Ω is a strict symmetric space with convex metric withrespect to the Thompson metric that satisfies the equivalentconditions (i), (ii) and (iii) of the Key Lemma.


Key Theorem II

Furthermore, (i) the order-reversing property of inversion,(ii) the harmonic-geometric-arithmetic mean inequality, and(iii) the Loewner-Heinz inequality all hold: for a, b ∈ Ω,

(i) b−1 ≤ a−1 if a ≤ b,

(ii) 2(a−1 + b−1)−1 ≤ a#b ≤ 12(a + b), and

(iii) at ≤ bt if a ≤ b, 0 ≤ t ≤ 1.


Key Theorem II

Furthermore, (i) the order-reversing property of inversion,(ii) the harmonic-geometric-arithmetic mean inequality, and(iii) the Loewner-Heinz inequality all hold: for a, b ∈ Ω,

(i) b−1 ≤ a−1 if a ≤ b,

(ii) 2(a−1 + b−1)−1 ≤ a#b ≤ 12(a + b), and

(iii) at ≤ bt if a ≤ b, 0 ≤ t ≤ 1.

Cones satisfying the hypotheses (and hence conclusions)of the preceding theorem will be called Loos symmetriccones.


Nonpositive Curvature

It is interesting to note that in the context of cones equippedwith symmetric structures the weakened form of thegeometric-arithmetic mean inequality

ε#x = x1/2 ≤ (ε + x)/2

yields both the Loewner-Heinz inequality and Busemannnonnegative curvature (in the sense that the Thompsonmetric is convex).

Question: Is convexity of the Thompson metric equivalentto convexity of the Hilbert metric?


Example: C∗-Algebras

If A is a unital C∗-algebra with identity e, then the cone A+

of positive invertible elements is an open normal convexcone in the closed subspace H(A) of hermitian elements.The set A

+ is a twisted subgroup (closed under(x, y) 7→ xy−1x) with unique square roots of themultiplicative group of invertible elements of A, hence apointed strict symmetric set with respect to x • y = xy−1xand distinguished point the identity e. Furthermore, thepowers computed in the algebra agree with those computedin (A+, •, e), so that the squaring map is continuous. Thecondition x1/2 ≤ (e + x)/2 is equivalent to (e − x)2 ≥ 0, thusvalid. Since Q(x)y = x(y−1)−1x = xyx, the mapping Q(x) isa continuous and linear on H(A).

It follows that the cone A+ is a Loos symmetric cone.


Harmonic-Arithmetic Approximation

Many results concerning the cone of positive elements inan algebra generalize to Loos symmetric cones.

For x, y in a Loos symmetric cone Ω in a Banach space V ,define H1 = H(x, y), the harmonic mean, and A1 = A(x, y),the arithmetic mean. Inductively define Hn+1 = H(Hn, An)and An+1 = A(Hn, An). Then for each n,

Hn ≤ Hn+1 ≤ x#y ≤ An+1 ≤ An,

and Hn → x#y, An → x#y.


Inequalities

Numerous operator inequalities remain valid in the moregeneral context of Loos symmetric cones, for example,

The Furuta Inequality: Let Ω be a Loos symmetric cone ina Banach space V and let 0 < b ≤ a. If 0 ≤ p, q, r ∈ R

satisfies p + 2r ≤ (1 + 2r)q and 1 ≤ q, then

bp+2r

q ≤ (br.ap)1

q .


JB-Algebras

A Jordan algebra is a vector space Z with a commutativemultiplication xy such that x(x2y) = x2(xy) holds for x, y ∈ Z.A JB-algebra V is a real Jordan algebra with unit eendowed with a complete norm || · || such that

||zw|| ≤ ||z|| ||w||, ||z2|| = ||z||2, ||z||2 ≤ ||z2 + w2||.

(The hermitian elements x = x∗ of any C∗-algebra form aJB-algebra with respect to the symmetric productx y := (xy + yx)/2. )


JB-Algebras

A Jordan algebra is a vector space Z with a commutativemultiplication xy such that x(x2y) = x2(xy) holds for x, y ∈ Z.A JB-algebra V is a real Jordan algebra with unit eendowed with a complete norm || · || such that

||zw|| ≤ ||z|| ||w||, ||z2|| = ||z||2, ||z||2 ≤ ||z2 + w2||.

(The hermitian elements x = x∗ of any C∗-algebra form aJB-algebra with respect to the symmetric productx y := (xy + yx)/2. )

Theorem. Let V be a JB-algebra and let Ω = exp V be theassociated positive cone. Then Ω is a Loos symmetric cone.


Hermitian Banach Algebras

Let Z be a unital Banach algebra Z with a continuousinvolution ∗ and let X consist of the self-adjoint elements ofZ. Let e denote the unit element of Z. The unital Banachalgebra Z is called hermitian if σ(x) ⊂ R and||x|| = sup |σ(x)| for every x = x∗. We let

Ω := x = x∗ : σ(x) ⊂ (0,∞).

Theorem. Let Z be a hermitian Banach algebra. Then Ω isa Loos symmetric cone of X.


Open Problem

The least squares mean A of positive matrices A1, . . . , An

minimizes

F (X) =n∑

i=1

d2(X,Ai),

where d is the trace Riemannian metric.

Problem. Does the least squares mean generalize to Loossymmetric cones?


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Axiomatic Symmetric Cones and the Thompson metricconstantin.vernicos.org/Articles/Hilbgeom.pdfThe Order For a normal cone Ω, the relation x ≤ y if and only if y −x ∈ Ω is a