Positive and Doubly Stochastic Maps, and Majorization in Euclidean Jordan Algebras

Home , Doubly stochastic matrix, Hermitian matrix

Linear Algebra and its Applications 528 (2017) 40–61

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Linear Algebra and its Applications

www.elsevier.com/locate/laa

Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras

M. Seetharama Gowda

Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA a r t i c l e i n f o a b s t r a c t

Article history: A positive map between Euclidean Jordan algebras is a Received 19 August 2015 (symmetric cone) order preserving linear map. We show that Accepted 17 February 2016 the norm of such a map is attained at the unit element, thus Available online 4 March 2016 obtaining an analog of the operator/matrix theoretic Russo– Submitted by P. Semrl Dye theorem. A doubly stochastic map between Euclidean This paper is dedicated, with much Jordan algebras is a positive, unital, and trace preserving map. admiration and appreciation, to We relate such maps to Jordan algebra automorphisms and Professor Rajendra Bhatia on the majorization. occasion of his 65th birthday © 2016 Elsevier Inc. All rights reserved. MSC: 15A18 15A48 15B51 15B57 17C30

Keywords: Euclidean Jordan algebra Positive map Doubly stochastic map Majorization

1. Introduction

This paper deals with positive and doubly stochastic maps, and majorization in Eu- clidean Jordan algebras. In the first part of the paper, we describe the spectral norm of a positive map between Euclidean Jordan algebras. To elaborate, consider the space Mn of all n × n complex matrices and its real subspace Hn of all n × n Hermitian matrices. A complex linear map Φ : Mn →Mk is said to be positive [5] if Φmaps positive semidefinite (Hermitian) matrices in Mn to positive semidefinite matrices in Mk. For such a map, the following result – a consequence of the so-called the Russo–Dye Theorem [5] – holds:

||Φ|| = ||Φ(I)||, where the norms are operator norms. Now, by writing ||A||∞ for the spectral norm/radius n n of A ∈H , observing ||A|| = ||A||∞, and restricting Φto H , the above relation yields

||Φ||∞ = ||Φ(I)||∞, (1)

n k where ||Φ||∞ is the operator norm of Φ :(H , || ·||∞) → (H , || ·||∞). In this paper, we show that (1) extends to any positive map between Euclidean Jordan algebras. A Euclidean Jordan algebra is a finite dimensional real inner product space with a compatible ‘Jordan product’. In such an algebra, the cone of squares induces a partial order with respect to which the positivity of a map is defined. Using the spectral decomposition of an element in such an algebra, one defines eigenvalues and the spectral norm. It is in this setting, a generalization of (1) is proved. We remark that unlike C∗-algebras which are associative, Euclidean Jordan algebras are non-associative. Along with the positivity result, we prove an analog of the following ‘spread inequality’ stated n k n for any two positive maps Φ1 and Φ2 from M to M and any x ∈H [6]:

|| − || ≤ ↓ − ↓ Φ1(x) Φ2(x) λ1(x) λn(x), (2)

↓ ↓ ∈Hn where λ1(x)and λn(x)are the largest and smallest eigenvalues of x . The second part of the paper deals with doubly stochastic maps and majorization. Given two vectors x and y in Rn, we say that x is majorized by y and write x ≺ y [16] if their decreasing rearrangements x↓ and y↓ satisfy

k k n n ↓ ≤ ↓ ↓ ↓ xi yi and xi = yi 1 1 1 1 for all 1 ≤ k ≤ n. When this happens, a theorem of Hardy, Littlewood, and Polya ([3], Theorem II.1.10) says that x = Dy, where D is a doubly stochastic matrix (which means that D is a nonnegative matrix with row and column sums equal to one). In 42 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61

addition, by a theorem of Birkhoﬀ ([3], Theorem II.2.3), there exist positive numbers λi and permutation matrices Pi such that

N N λi =1 and x = λi Pi y. (3) 1 1

Now for given Hermitian matrices X and Y in Hn, one says that X is majorized by Y (see [1], page 234) if λ↓(X) ≺ λ↓(Y ), where λ↓(X) denotes the vector of eigenvalues of X written in the decreasing order, etc. In this case, it is known, (see e.g., [1], Theorem 7.1) X =Φ(Y )for some linear map Φ : Hn →Hn (equivalently, Φ : Mn →Mn) that is doubly stochastic (which means that Φis positive, unital, and trace preserving, see

Section 2 for deﬁnitions). Moreover, there exist positive numbers λi and unitary matrices Ui such that

N N ∗ λi =1 and X = λi Ui YUi. (4) 1 1

With the observations that Rn and Hn are two instances of Euclidean Jordan algebras and permutation matrices on Rn and maps of the form X → U ∗XU (with unitary U) on Hn are Jordan algebra automorphisms (these are invertible linear maps that preserve the Jordan product), we extend the above two results to Euclidean Jordan algebras. By deﬁning x ≺ y to mean λ↓(x) ≺ λ↓(y), we show that in any Euclidean Jordan algebra V , the convex hull of the (algebra) automorphism group Aut(V )and the set of all doubly stochastic maps DS(V) are related by

conv (Aut(V)) y ⊆ DS(V) y ⊆{x ∈ V : x ≺ y} (y ∈ V ), with equalities when V is a simple Euclidean Jordan algebra. In addition, we show that for the Jordan spin algebra Ln,

conv (Aut(Ln)) = DS(Ln). (5)

An outline of the paper is as follows. In Section 2, we cover some preliminary material. Section 3 deals with a consequence of Hirzebruch’s min-max theorem and the spectral norm. In Section 4, we prove analogs of (1) and (2) for Euclidean Jordan algebras. Examples of doubly stochastic maps on Euclidean Jordan algebras are given in Section 5. Majorization results are covered in Section 6. In Sections 7 and 8, we relate Jordan algebra automorphisms and doubly stochastic maps.

2. Preliminaries

Throughout this paper, V and V denote Euclidean Jordan algebras. Recall that a Euclidean Jordan algebra V is a ﬁnite dimensional real vector space that carries an inner M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 43 product x, y (with induced norm ||x||) and a bilinear Jordan product x ◦ y satisfying the following properties:

(a) x ◦ y = y ◦ x, (b) x ◦ (x2 ◦ y) = x2 ◦ (x ◦ y), where x2 = x ◦ x, (c) x ◦ y, z = x, y ◦ z .

It is known that every Euclidean Jordan algebra has a ‘unit’ element e that satisﬁes x ◦ e = x for all x. The monograph [8] is our primary source on Euclidean Jordan algebras; for a short summery relevant to our study, see [10]. Clearly, Rn, with the usual inner product and componentwise product, is a Euclidean Jordan algebra. It is well-known that every nonzero Euclidean Jordan algebra is isomorphic to a product of the following simple algebras:

(i) the algebra Sn of all n × n real symmetric matrices, (ii) the algebra Hn of n × n complex Hermitian matrices, (iii) the algebra Qn of n × n quaternion Hermitian matrices, (iv) the algebra O3 of 3 × 3octonion Hermitian matrices, and (v) the Jordan spin algebra Ln, n ≥ 3.

In all the matrix algebras above, the Jordan product and inner product are respectively given by

XY + YX X ◦ Y = and X, Y = Re tr (XY ), 2 where the trace of a matrix is the sum of its diagonal elements. The Jordan spin algebra Ln is deﬁned as follows. For n > 1, consider Rn with the usual inner product. Writing any element x ∈ Rn as x x = 0 x

n−1 with x0 ∈ R and x ∈ R , we deﬁne the Jordan product: x y x, y x ◦ y = 0 ◦ 0 := . (6) x y x0y + y0x

Then Ln denotes the Euclidean Jordan algebra (Rn, ◦, ·, · ).

In a Euclidean Jordan algebra V , we let

V+ = {x ◦ x : x ∈ V } 44 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 denote its symmetric cone. This closed convex cone is self-dual and homogeneous. In particular,

x ∈ V+ ⇔x, y ≥0 ∀ y ∈ V+. (7)

In V , we write

x ≥ y (or y ≤ x)whenx − y ∈ V+.

An element c in V is an idempotent if c2 = c; it is a primitive idempotent if it is nonzero and cannot be written as a sum of two nonzero idempotents. The set of all primitive idempotents in V , denoted by τ(V ), is a compact set, see [8], page 78. For each c ∈ τ(V ), we let c c c¯ := (= ). e, c ||c||2

A Jordan frame in V is a set {e1, e2, ..., er}, where each ei is a primitive idempotent, ei ◦ ej =0for all i = j, and e = e1 + e2 + ···+ er. In V , any two Jordan frames will have the same number of objects; this number is the rank of V . Throughout this paper, we let V and V denote Euclidean Jordan algebras with respective symmetric cones V+ and V+, and respective unit elements e and e . Ranks of V and V are denoted by r and s respectively. Any element x ∈ V will have a spectral decomposition

x = x1e1 + x2e2 + ···+ xrer, (8) where {e1, e2, ..., er} is a Jordan frame and the real numbers x1, x2, ..., xr are (called) the ‘eigenvalues’ of x. As ei, ej =0for all i = j, we note that

x, ei x, ei xi := = = x, e¯i ,i=1, 2,...,r. ei,ei e, ei

r ↓ We deﬁne λ(x)to be any vector in R with components x1, x2, ..., xr and λ (x)be its (unique) decreasing rearrangement. In particular, we write

↓ { } ↓ { } λ1(x):=max x1,x2,...,xr and λr(x):=min x1,x2,...,xr .

The trace and the spectral norm of x are respectively deﬁned by

r tr(x):= xi and ||x||∞ := max |xi|. 1≤i≤r 1

Then

x, y tr := tr(x ◦ y) M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 45 deﬁnes the canonical inner product on V which is also compatible with the Jordan product on V . Various concepts/results/decompositions remain the same when the given inner product is replaced by the canonical inner product; in particular, for an element, the spectral decomposition, eigenvalues, and trace remain the same. We note that under the canonical inner product, the norm of any element x, via (8), is r 2 ||x||2 := x, x tr = |xi| . 1

In particular, in this setting, the norm of a primitive idempotent is one and tr(x) =

x, e tr . When V is simple, the given inner product is a (positive) multiple of the canonical inner product, see Prop. III.4.1 in [8].

Given Euclidean Jordan algebras V and V , we let B(V, V )denote the set of all (continuous) linear maps from V to V . Note that B(V, V )is a (ﬁnite dimensional) normed linear space with respect to the operator norm. We recall the following. An invertible linear transformation Γ ∈B(V, V )is

(i) a (Jordan) algebra automorphism if

Γ(x ◦ y)=Γ(x) ◦ Γ(y) ∀ x, y ∈ V ;

(ii) a (symmetric) cone automorphism if

Γ(V+)=V+;

(iii) orthogonal if Γ(x), Γ(y) = x, y for all x, y ∈ V .

We recall that a complex linear map Φ : Mn →Mk is doubly stochastic [1] if it is positive, unital (which means, Φmaps the identity matrix in Mn to identity matrix in Mk), and trace preserving (that is, tr(Φ(X)) = tr(X)for all X ∈Mn). We have an analogous deﬁnition for Euclidean Jordan algebras. A linear map Φ : V → V is said to be

⊆ (a) positive if Φ(V+) V+, (b) unital if Φ(e) = e, (c) trace preserving if tr(Φ(x)) = tr(x)for all x ∈ V , and (d) doubly stochastic if it is positive, unital, and trace preserving.

Recall that the transpose of a linear map Φ : V → V is the map ΦT : V → V deﬁned by: Φ(x), y = x, ΦT (y) for all x ∈ V and y ∈ V . We have the following. 46 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61

(i) The positivity of Φ : V → V is equivalent to that of ΦT : V → V . (This is an easy consequence of (7)). (ii) When V and V carry canonical inner products, the trace preserving (unital) property of Φis equivalent to the unital (respectively, trace preserving) property of ΦT . This can be seen by observing

x, e = tr(x)=tr(Φ(x)) = Φ(x),e = x, ΦT (e) . (9)

In particular, when V is simple and Φ ∈B(V, V ), Φis doubly stochastic if and only if its transpose is doubly stochastic.

We respectively write Aut(V ), Aut(V+), Orth(V ), and DS(V )for the set of all algebra automorphisms, cone automorphisms, orthogonal maps, and doubly stochastic maps. It is easy to see that algebra automorphisms of Rn are just permutation matrices. As noted in [10], algebra automorphisms of Hn (of Sn) are given by Φ(X) = U ∗XU for some unitary (respectively, real orthogonal) matrix U and those of Ln are given by matrices of the form 10 Ψ= , 0 D for some orthogonal matrix D on Rn−1.

Let V1 and V2 be two Euclidean Jordan algebras. We deﬁne the Jordan and inner products on V1 × V2 by (x1, x2) ◦ (y1, y2) =(x1 ◦ y1, x2 ◦ y2)and (x1, x2), (y1, y2) = x1, y1 + x2, y2 . If Φi ∈ Aut(Vi)for i =1, 2, then the map (Φ1, Φ2) :(x1, x2) → (Φ(x1), Φ(x2)) is in Aut(V1 × V2). In the following result (which is perhaps known), we state a converse and sketch a proof.

Proposition 1. Let V1 and V2 be two non-isomorphic simple Euclidean Jordan algebras with Φ ∈ Aut(V1 × V2). Then Φ is of the form (Φ1, Φ2) for some Φi ∈ Aut(Vi) for i =1, 2.

Proof. We assume without loss of generality that V1 and V2 carry canonical inner products. To simplify the notation, we write Φin the (block) form: AB Φ= , CD where A : V1 → V1, B : V2 → V1, C : V1 → V2, and D : V2 → V2 are linear transformations. We show that A and D are algebra automorphisms, and B and C are zero. Now, for xi and yi in Vi, M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 47

A(x1 ◦ y1)+B(x2 ◦ y2)=(Ax1 + Bx2) ◦ (Ay1 + By2)

C(x1 ◦ y1)+D(x2 ◦ y2)=(Cx1 + Dx2) ◦ (Cy1 + Dy2).

By choosing appropriate xi and yi, we see that A, B, C, D are algebra homomorphisms (that is, they preserve Jordan products) and Ax1 ◦ By2 =0and Dy2 ◦ Cx1 =0. As the kernels of these four transformations are ideals and Vi s are assumed to be simple (which means that they do not contain any non-trivial ideals, see page 51, [8]), we see that A and D are either zero or invertible, B and C are either zero or one-to-one. In addition, T T from Ax1, By2 =0and Dy2, Cx1 =0, we get A B =0and D C =0. The cases A =0, B =0and C =0and D =0are not possible due to the invertibility of Φ. The cases A =0, B =0and C =0, D =0are also not possible. Now suppose A =0, B =0 and D =0, C =0. Then, as B : V2 → V1 and C : V1 → V2 are one-to-one, the algebras V1 and V2 have equal dimensions, and B (which is an algebra homomorphism) becomes an algebra isomorphism. Thus, B maps Jordan frames to Jordan frames and (hence) preserves trace and (canonical) inner product. In this case, V1 and V2 are isomorphic, contradicting our assumption. Hence, this situation does not arise. In the (last) case when A =0, B =0and C =0, D =0, we have A ∈ Aut(V1)and D ∈ Aut(V2). 2

3. The spectral norm

Theorem 1. (See Hirzebruch [14].) Let V be a Euclidean Jordan algebra of rank r. For any element x ∈ V , the smallest and largest eigenvalues are given, respectively, by

↓ ↓ λr(x)= min x, c¯ and λ1(x)= max x, c¯ . c∈τ(V ) c∈τ(V )

The above result is usually stated for simple Euclidean Jordan algebras [14,18]. To see the general case, we write V as a product of simple algebras: V = V1 × V2 ×···×Vk. Then, any primitive idempotent in V is of the form (0, 0, ..., 0, ei, 0, 0 ..., 0) for some primitive idempotent ei in Vi (i =1, 2, ..., k). Also, for any x in V , eigenvalues of x come from the eigenvalues of the components of x. With these facts, one easily veriﬁes the above result.

Corollary 1. Let x, a ∈ V with a ≥ 0. Then,

(i) maxc∈τ(V ) |x, c¯ | =maxc∈τ(V )|x|, c¯ = ||x||∞. (ii) −a ≤ x ≤ a ⇒||x||∞ ≤||a||∞.

≥ || || ↓ Proof. (i)For any y 0in V , by the above theorem, y ∞ =λ1(y) =maxc∈τ(V ) y, c¯ . ∈ r | | Now, consider any x V with its spectral decomposition x = 1 xiei. From y := x = r | | ± ≥ ∈ ± ≥ 1 xi ei, we see that y x 0and so for any c τ(V ), y x, c¯ 0, that is, |x, c¯ | ≤y, c¯ . Since ||y||∞ = ||x||∞,

max |x, c¯ | ≤ max y, c¯ = ||y||∞ = ||x||∞. c∈τ(V ) c∈τ(V ) 48 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61

However, |xi| = |x, e¯i | ≤ maxc∈τ(V ) |x, c¯ | for all i, and so, ||x||∞ ≤ maxc∈τ(V ) |x, c¯ |. Thus we have (i). (ii)For any c ∈ τ(V ), from −a ≤ x ≤ a ⇒−a, c¯ ≤x, c¯ ≤a, c¯ we get

|x, c¯ | ≤ a, c¯ .

Taking the maximum over τ(V ), using (i), we get ||x||∞ ≤||a||∞. 2

From Item (i)of the corollary, we get the triangle inequality

||x + y||∞ ≤||x||∞ + ||y||∞.

Thus, || ·||∞ is a norm on V , a fact that is already known, see Theorem 4.1, [20].

4. The norm of a positive map

For a linear map Φ : V → V , we deﬁne

||Φ||∞ := max ||Φ(x)||∞. x∈V, ||x||∞≤1

Theorem 2. Let Φ : V → V be a positive map. Then

||Φ||∞ = ||Φ(e)||∞. ∈ r Proof. For any x V , by writing the spectral decomposition x = 1 xiei and using the r −|| || ≤ ≤|| || fact that e = 1 ei, we get x ∞ e x x ∞ e. From the linearity and positivity of Φ, we get −||x||∞ Φ(e) ≤ Φ(x) ≤||x||∞ Φ(e). Now, Item (ii)in Corollary 1 yields ||Φ(x)||∞ ≤||x||∞ ||Φ(e)||∞. Hence, ||Φ||∞ ≤||Φ(e)||∞. As ||e||∞ =1, we have ||Φ||∞ = ||Φ(e)||∞. 2

Corollary 2. Suppose φ : V → R is a linear functional on V . Then ||φ||∞ = φ(e) if and only if φ is positive.

Proof. Assume ﬁrst that φ is positive. Deﬁne Φ : V → V by Φ(x) = φ(x) e. Then Φis positive on V . An application of the above theorem gives the equality ||φ||∞ = φ(e). To see the converse, let ||φ||∞ = φ(e). Without loss of generality, let φ(e) =1. For any x ∈ V with eigenvalues x1, x2, ..., xr,

|φ(x) − λ| = |φ(x − λe)|≤||φ||∞ ||x − λe||∞ =max|xi − λ| i for all λ ∈ R. This implies that φ(x)is between min xi and max xi. In particular, taking x ≥ 0(for which min xi ≥ 0), we see that φ(x) ≥ 0. 2 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 49

Theorem 3. Suppose Φ : V → V is a linear map with ||Φ||∞ =1and Φ(e) = e . Then Φ is positive.

Proof. Without loss of generality, we assume that V carries the canonical inner product. We ﬁx c ∈ τ(V )and deﬁne

φ : V → R, φ(x)=Φ(x),c (x ∈ V ).

Then, φ is a linear functional with φ(e) = Φ(e), c = e, c =1. (We note that as V carries the canonical inner product, the norm of any primitive element is one.) In addition,

|φ(x)| = |Φ(x),c | ≤ ||Φ(x)||∞ ≤||x||∞, where we have used Item (i)of Corollary 1 for Φ(x)in V . We see that conditions of the previous corollary hold. Hence, φ(x) ≥ 0for all x ≥ 0in V . This means that when x ≥ 0, Φ(x), c ≥ 0for all c ∈ τ(V ). This leads to, by considering nonnegative linear ≥ ∈ combinations of elements of τ(V ), Φ(x), y 0for all y V+. By an application of (7), we get Φ(x) ≥ 0for any x ≥ 0. This proves that Φis a positive map. 2

We illustrate the above results by some examples.

Example 1. Let b ∈ V . Then the quadratic representation Pb is deﬁned by

2 Pb(x)=2b ◦ (b ◦ x) − b ◦ x.

Its norm is given by

2 ||Pb||∞ = ||b||∞.

This follows easily from the observations that Pb is a positive map (see Prop. III.2.2, 2 2 2 [8]), Pb(e) = b and ||b ||∞ = ||b||∞.

Example 2. Let L : V → V be a Z-transformation [11], that is, L is linear and satisﬁes

x ≥ 0,y ≥ 0, and x, y =0⇒L(x),y ≤0.

If L is positive stable (which means that all eigenvalues of L have positive real parts), then L−1 is a positive map ([11], Theorem 6). Hence,

−1 −1 ||L ||∞ = ||L (e)||∞.

We can further specialize by taking V = Sn. Then for any A ∈ Rn×n, the transformations

T T n LA(X):=AX + XA and SA(X):=X − AXA (X ∈S ) 50 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61

(which are respectively, the Lyapunov and Stein transformations) are Z-transformations n on S . When they are positive stable (which happens when A is positive stable for LA −1 −1 and Schur stable for SA) we see that the spectral norms of LA and SA are attained at the Identity matrix in Sn. These results are known, see e.g., [4,5].

{ } Example 3. Corresponding to a Jordan frame e1, e2, ..., er in V , consider the Peirce { ∈ ◦ } decomposition V = i≤j Vij [8], where Vii := x V : x ei = x and for i = j, { ∈ ◦ 1 ◦ } ∈ Vij := x V : x ei = 2 x = x ej . Thus, for any x V , we can write x = xij, where xij ∈ Vij. (10) i≤j

As Vii = Rei for all i, we may write xii = xiei with xi ∈ R. Then, the above decomposition reads:

r x = xiei + xij. (11) 1 i

As Vij s are mutually orthogonal, tr(x) = x1 + x2 + ···+ xr. Also, when x ∈ V+, xi ≥ 0 for all i.

Let A =[aij]be any r ×r real symmetric matrix. Given (10), we can deﬁne the ‘Schur product’ of A and x in V by A • x := aij xij. (12) i≤j

Now suppose that A is positive semideﬁnite and let Φ(x) := A • x. It has been shown in [13], Proposition 2.2, that in this situation, x ∈ V+ ⇒ Φ(x) ∈ V+. This means that Φ is a positive map. Then,

||Φ||∞ = ||Φ(e)||∞ = ||a11e1 + a22e2 + ···+ arrer||∞ =max|aii|. 1≤i≤r

The following result is motivated by Theorem 1 in [6].

Theorem 4. Let Φ1 and Φ2 be two positive unital maps from V to V . Then for any x ∈ V ,

|| − || ≤ ↓ − ↓ Φ1(x) Φ2(x) ∞ λ1(x) λr(x).

↓ ↓ Proof. Fix x in V , and let (for ease of notation), α = λr(x)and β = λ1(x). Then,

αe≤ x ≤ βe. M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 51

Using the positivity and the unital properties of Φi (i =1, 2), we get αe ≤ Φ1(x) ≤ βe and − βe ≤−Φ2(x) ≤−αe.

Hence,

(α − β) e ≤ Φ1(x) − Φ2(x) ≤ (β − α)e .

By Item (ii)in Corollary 1, we get the required inequality. 2

5. Doubly stochastic maps

Recall that a linear map Φon a Euclidean Jordan algebra V is doubly stochastic if it is positive, unital, and trace preserving. It is easy to see that the set of all such maps, DS(V ), is convex, and (from Theorem 2) compact in B(V, V ). We now list some examples.

Example 4. Consider a doubly stochastic map Φ : Mn →Mn. By definition, this map takes positive semidefinite matrices in Mn to positive semidefinite matrices and is unital and trace preserving. As every Hermitian matrix is the difference of two positive semidefinite matrices, we see that the restriction of Φto Hn is a doubly stochastic map on the Euclidean Jordan algebra Hn. Conversely, as

Mn = {A + iB : A, B ∈Hn}, it follows that every (real linear) doubly stochastic map Φon Hn can be extended to a (complex linear) doubly stochastic map Φon Mn by

Φ( A + iB)=Φ(A)+iΦ(B).

Example 5. Every Jordan algebra automorphism (on a Euclidean Jordan algebra) is a doubly stochastic map. This is because any Jordan algebra automorphism Φsatisﬁes Φ(x2) =Φ(x)2 (implying positivity), Φ(e) = e (unital property), and maps Jordan frames to Jordan frames (implying the trace preserving property). Thus, Aut(V ) ⊆ DS(V ); Denoting the convex hull of a set S by conv(S), we see that conv(Aut(V )) is convex, and

conv(Aut(V )) ⊆ DS(V ). (13)

We shall see later that Aut(V )is also compact in B(V, V ).

Example 6. Let V be of rank r. It is easy to see that the map

tr(x) Φ(x):= e, r is doubly stochastic. 52 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61

Example 7. We ﬁx a Jordan frame {e1, e2, ..., er} in V and consider the Peirce decomposition (11). Then the ‘diagonal map’ Diag : V → V deﬁned by

Diag(x)=x1e1 + x2e2 + ···+ xrer (x ∈ V ) (14) is doubly stochastic.

Example 8. Consider a Jordan frame {e1, e2, ..., er} in V and the set up of Example 3. Let A be an r × r real symmetric positive semideﬁnite matrix with every diagonal entry one. Then, the map

Φ(x):=A • x is doubly stochastic.

Example 9. We ﬁx a nonzero idempotent c = e in V , and let

V (c, γ):={x ∈ V : x ◦ c = γx},

∈{ 1 } where γ 0, 2 , 1 . Then the Peirce (orthogonal) decomposition

1 V = V (c, 1) + V (c, )+V (c, 0) (15) 2 holds [8]. In particular, for any element x ∈ V , we have

1 x = u + v + w with u ∈ V (c, 1), v ∈ V (c, ), w ∈ V (c, 0). (16) 2 Now, consider the ‘pinching map’ Φon V deﬁned by

Φ(x):=u + w. (17)

We claim that Φ is doubly stochastic. To see this, let ω = c − (e − c). By writing the spectral decomposition c = e1 + e2 + ···+ ek +0 ek+1 + ···+0 er, 1 ≤ k

ω = e1 + e2 + ···+ ek − (ek+1 + ···+ er).

2 As ω = e, by Prop. II.4.4 in [8], Pω is an algebra automorphism of V . Using the deﬁnition of Pω and properties of the spaces V (c, γ), one veriﬁes that

x + P (x) ω = u + w. (18) 2 This means that the pinching map is the average of two algebra automorphisms, namely the identity transformation and Pω. Thus, it is doubly stochastic. M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 53

The following result exhibits a connection between doubly stochastic maps and doubly stochastic matrices.

Theorem 5. Let Φ ∈B(V, V ) be doubly stochastic. Let {e1, e2, ..., er} be a Jordan frame in V and {f1, f2, ..., fs} be a Jordan frame in V . Then the matrix A =[aij], aij := Φ(ei), fj tr , is a doubly stochastic matrix.

Proof. We assume without loss of generality that both V and V carry canonical inner products. With respect to these inner products, Φcontinues to have the doubly stochastic property. In particular, as observed previously in (9), the trace preserving property of Φ becomes the unital property of the transpose ΦT . ⊆ Now, Φ(V+) V+ together with (7) implies that A is a nonnegative matrix. From Φ(e) = e and the orthogonality relations of a Jordan frame, we have, for each ﬁxed j,

r aij = Φ(e),fj = e ,fj = fj,fj =1. i=1

As ΦT (e) = e, for each ﬁxed i,

s T aij = Φ(ei),e = ei, Φ (e ) = ei,e = ei,ei =1. j=1

Thus A is doubly stochastic. 2

Remarks. Suppose for a given map Φ ∈B(V, V ), the matrix [Φ(ei), fj tr ]is doubly stochastic for arbitrary Jordan frames in V and V . Then, an easy argument shows that Φis doubly stochastic. We omit the proof.

6. Majorization

Given x, y ∈ V , we say that x is majorized by y and write x ≺ y if the eigenvalue vectors of x and y satisfy

λ(x) ≺ λ(y).

Theorem 6. For x, y ∈ V , consider the following statements:

(a) x =Ψ(y), where Ψ is a convex combination of algebra automorphisms of V . (b) x =Φ(y), where Φ is doubly stochastic on V . (c) x ≺ y.

Then, (a) ⇒ (b) ⇒ (c). Furthermore, when V is Rn or simple, the reverse implications hold. 54 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61

Proof. (a) ⇒ (b): This follows from (13). ⇒ (b) (c): Let x =Φ(y), where Φis doubly stochastic. Writing the spectral decomposi- r r tions x = 1 xiei and y = 1 yifi, we see that λ(x) = A λ(y), where A =[Φ(fj), ei tr ], T λ(x) =[x1, x2, ..., xr] , etc. As A is a doubly stochastic matrix by Theorem 5, we see that λ(x) ≺ λ(y) proving (c). n ⇒ Now suppose (c)holds. When V is R , the implication (c) (a)is covered by (3). ≺ r Now assume that V is simple. Let x y with spectral decompositions x = 1 xiei and r ≥ ≥···≥ ≥ ≥···≥ y = 1 yifi; without loss of generality, let x1 x2 xr and y1 y2 yr so ↓ T ↓ T ↓ ≺ ↓ that λ (x) =[x1, x2, ..., xr] and λ (y) =[y1, y2, ..., yr] . By deﬁnition, λ (x) λ (y). N Then there exists positive numbers μi and permutation matrices Pi such that 1 μi =1 and

N ↓ ↓ λ (x)= μjPj(λ (y)). j=1

Now, each P induces a permutation σ on {1, 2, ..., r}. Let (P (λ↓(y))) = y . Then j j j i σj (i) N xi = j=1 μjyσj (i). Hence, ⎛ ⎞ r r N N r ⎝ ⎠ x = xiei = μjyσj (i) ei = μj yσj (i)ei . i=1 i=1 j=1 j=1 i=1

−1 ∈ −1 Now for each j =1, 2, ..., N, let Ψj Aut(V )such that Ψj (ei) = fσj (i) for all i. Since

V is simple, such an automorphism exists, see [8], Theorem IV.2.5. Then ei =Ψj(fσj (i)) for all i. Hence, N r N r N

x = μj yσj (i)ei = μjΨj yσj (i)fσj (i) = μjΨj (y) j=1 i=1 j=1 i=1 j=1 r r 2 as y = 1 yifi = i=1 yσj (i)fσj (i) for every j. We thus have (a).

Here are some illustrations of the above theorem.

• From Example 7, Diag(x) ≺ x. This is a generalization of the Schur inequality, see [12]. • When A is an r × r real symmetric positive semideﬁnite matrix with every diagonal entry one, by Example 8, A • x ≺ x. This yields λ(A • x) ≺ λ(x), extending a result of Bapat and Sunder, see [2], Corollary 2. • From Example 9, for the pinching map (17), u + w ≺ x. This result is proved in [20] for simple algebras, based on case by case analysis.

Remarks. Let W be a ﬁnite dimensional real inner product space and G be a closed subgroup of the group of orthogonal (that is, inner product preserving) linear maps on W . M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 55

In several works, see for example, [7,9,17,19], the concept of G-majorization is developed and studied. One says that x is G-majorized by y in W if x ∈ conv(G) y (that is, x is in the convex hull of the G-orbit of y). If we take W to be a Euclidean Jordan algebra with the canonical inner product, then G =Aut(W ) ⊆ Orth(W )(see [8], page 57). In this setting, the implication (a) ⇒ (c)in Theorem 6 shows that G-majorization is stronger than our eigenvalue-majorization. While they are the same when the algebra is simple, they could be diﬀerent in the general case, as our next example shows.

Example 10. We recall that S2 is the algebra of all real 2 × 2 symmetric matrices, with ◦ XY +YX ×S2 ◦ ◦ X Y = 2 and X, Y = tr(XY ). Let V = R with (λ, X) (μ, Y ) := (λμ, X Y ) and (λ, X), (μ, Y ) := λμ + X, Y for λ, μ ∈ R and X, Y ∈S2. Consider two elements of V : p =(0, A)and q =(1, B), where 10 00 A = and B = . 00 00

Clearly, λ↓(p) = λ↓(q)and so p ≺ q. Now, it follows from Proposition 1 that algebra automorphisms of V are given by (λ, X) → (λ, U T XU)for some orthogonal matrix U. Thus, p =Ψ( q)for any Ψwhich is a convex combination of algebra automorphisms of V . So in this algebra, the implication (c) ⇒ (a)does not hold. We also observe that with G =Aut(V ), p is not G-majorized by q. Now, consider the map Φ deﬁned by xy λ 0 Φ: λ, → x, . yz 0 z

It is easy to see that Φis doubly stochastic. Also, p =Φ(q). This means that in this algebra,

conv(Aut(V )) =DS( V ).

7. Relations between Aut(V ), Aut(V+ ) and DS(V )

Theorem 7. The following statements are equivalent:

(a) Φ ∈ Aut(V ). (b) Φ ∈ DS(V ) and ||Φ(x)||∞ = ||x||∞ for all x ∈ V .

Proof. We assume without loss of generality that V carries the canonical inner product. (a) ⇒ (b): Let Φ ∈ Aut(V ). Clearly, Φis doubly stochastic and maps Jordan frames to Jordan frames. In particular, it preserves eigenvalues and so, ||Φ(x)||∞ = ||x||∞ for all x ∈ V . Thus, we have (b). 56 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61

Now assume that (b)holds. We show the following:

(i)Φc) ∈ τ(V )for all c ∈ τ(V ), where τ(V ) denotes the set of all primitive idempotents in V ; ( ii)Φmaps Jordan frames to Jordan frames; ( iii) Φ preserves eigenvalues;

(iv)Φ ∈ Aut(V+).

To see (i), let c ∈ τ(V ). Then (from the positive property) Φ(c) ∈ V+ and ||Φ(c)||∞ = ||c||∞ =1. So, in the spectral decomposition Φ(c) = λ1f1 + λ2f2 + ···+ λrfr, every λi is nonnegative and maxi λi =1. Without loss of generality, let λ1 =1so that Φ(c) = r r f1 + 2 λifi. Since Φ preserves the trace, 1 = tr(c) = tr(Φ(c)) =1 + 2 λi. Thus, r ≥ ∈ 2 λi =0and λi =0for all i 2. Therefore, Φ(c) = f1 τ(V ). Now, to see (ii), let {e1, e2 ..., er} be any Jordan frame in V . For any i, put Φ(ei) = pi. Then, Φ(e) = e implies that p1 +p2 +···+pr = e and so, p1 ◦ p1 +p1 ◦ p2 +···+p1 ◦ pr = p1 ◦ e = p1. As p1 ◦ p1 = p1 (from (i)), this leads to

p1 ◦ p2 + ···+ p1 ◦ pr =0. r Taking the trace, we get 2 p1, pi = 0 (recall that V carries the canonical inner product). As pi ∈ τ(V ), from (7), we get p1, pi =0for all i ≥ 2. From this and Prop. 6 in [10], we infer that p1 ◦ pi =0for all i ≥ 2. In a similar way, we show that pi ◦ pj =0for { } all i = j. Thus, p1, p2, ..., pr is a Jordan frame. So, (ii)holds. r r Now, from the spectral decomposition x = 1 xiei, we get Φ(x) = 1 xi Φ(ei). From || ||2 r 2 (ii), we see that the eigenvalues of x and Φ(x)are the same. Thus, Φ(x) 2 = 1 xi = || ||2 x 2, proving the second part of (c)as well as the invertibility of Φ. As any x ∈ V+ is a ﬁnite nonnegative linear combination of elements of τ(V )and Φ ⊆ −1 preserves τ(V ), we see that Φ(V+) V+. We now show that Φ also preserves V+. Let ∈ −1 r c τ(V )and p := Φ (c). Starting with the spectral decomposition p = 1 αiei, via (ii), r we get the spectral decomposition c =Φ(p) = 1 αi Φ(ei). Thus, αis are the eigenvalues of c; we may let, without loss of generality, α1 =1and αi =0for all i ≥ 2. This means −1 that p = e1 and so Φ c = e1 ∈ τ(V ). Once again, by considering nonnegative ﬁnite −1 linear combinations of elements of τ(V ), we get Φ (V+) ⊆ V+. Combining this with Φ(V+) ⊆ V+, we get Φ ∈ Aut(V+). Thus we have (c). (c) ⇒ (a): The second part of condition (c)says that Φis an isometry (hence orthogonal) under the canonical inner product. So condition (c)means that Φ ∈ Aut(V+) ∩ Orth(V ). As V carries the canonical inner product, Φ ∈ Aut(V+) ∩ Orth(V ) =Aut(V ), see [8], page 57. Thus, we have (a). 2

Theorem 8. Suppose V is simple. Then

Aut(V )=Aut(V+) ∩ DS(V )={Φ ∈ Aut(V+):Φ(e)=e}. M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 57

Proof. The inclusions Aut(V ) ⊆ Aut(V+) ∩ DS(V ) ⊆{Φ ∈ Aut(V+) :Φ(e) = e} always hold. When V is simple, the reverse inclusions follow from

{Φ ∈ Aut(V+):Φ(e)=e} =Aut(V+) ∩ Orth(V )=Aut(V ), see [8], pages 56 and 57, Propositions III.4.3 and I.4.3. 2

Remarks. It follows from Theorem 7 that Aut(V )is compact in B(V, V ). Thus, conv(Aut(V )) is a compact convex subset of DS(V ). When V is simple, the equivalence of (a) and (b) in Theorem 6 shows that pointwise equality

conv (Aut(V)) z =DS(V)z (z ∈ V ) holds. Example 10 shows that such an equality may not hold in non-simple algebras. We now ask if a stronger equality conv (Aut(V)) =DS(V)can hold in simple algebras. The following result and the example shows that the equality holds in the algebra Ln (which is of rank 2), but may fail in other algebras.

Theorem 9. Every doubly stochastic map on Ln is a convex combination of algebra automorphisms, that is, it is of the form

N A = λi Ψi, 1 N where λis are positive, 1 λi =1, and 10 Ψi = , 0 Ui

n−1 for some orthogonal matrix Ui on R . Thus,

conv(Aut(Ln)) = DS(Ln).

Proof. We recall that for n > 1, Ln =(Rn, ◦, ·, · ), where Rn carries the usual inner product and the Jordan product is given by (6). The vector e in Rn with one in the n T ﬁrst slot and zeros elsewhere is the unit element of L . Also, a vector x =[x0, x] ∈ × n−1 Ln Ln ≥|||| R R belongs to + (the symmetric cone of ) if and only if x0 x 2. As n observed previously, algebra automorphisms of L have the form of (above) Ψi and so a convex combination of such automorphisms is doubly stochastic. Now consider a doubly stochastic map A on Ln written in the matrix form: abT A = , cD 58 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 where a ∈ R, b, c ∈ Rn−1 and D ∈ R(n−1)×(n−1). From Ae = e, we get a =1and c =0. As Ln is an algebra where the inner product is a multiple of the trace inner product, the trace preserving property of A is equivalent to AT e = e. This implies that b =0. Thus, 10 A = . (19) 0 D

We now show that D is a convex combination of orthogonal matrices. Let u be a vector n−1 T ∈Ln ∈Ln of norm one in R . As x =[1 u] +, we must have Ax +. This implies that ||Du|| ≤ 1. Hence, ||D|| ≤ 1. (Thus, every doubly stochastic matrix on Ln is of the form (19) for some ‘contractive’ matrix D.) Now, using the singular value decomposition (for the real matrix D), we can write

D = Udiag(s1,s2,...,sn−1)W, where U and W are (real) orthogonal matrices. As ||D|| ≤ 1, we have 0 ≤ si ≤ 1 for all i. Thus, we can write diag(s1, s2, ..., sn−1)as a convex combination of diagonal matrices Di, each of the form diag(ε1, ε2, ..., εn−1), with |εj | =1for all j. We see that D is a convex combination of orthogonal matrices of the form Ui := UDiW . Using Ui, we deﬁne the algebra automorphism Ψi. It follows that A is a convex combination of these algebra automorphisms. 2

Example 11. Consider the simple algebra H3 of all 3 × 3complex Hermitian matrices. We show that

conv(Aut(H3)) =DS( H3) (20) by using an example of Landau and Streater ([15], page 118) given in connection with the extreme points of the set of all doubly stochastic maps on M3. Consider the map Φ : H3 →H3 deﬁned by

∗ ∗ ∗ ∈H3 Φ(X):=V1 XV1 + V2 XV2 + V3 XV3 (X ), where V = √1 J , i 2 i ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 −i 0 00 0 00i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ J1 = ⎣ i 00⎦ ,J2 = ⎣ 00−i ⎦ ,J3 = ⎣ 000⎦ . 000 0 i 0 −i 00

2 2 2 ∈ H3 As each Vi is Hermitian and V1 + V2 + V3 = I, we see that Φ DS( ). We now claim that Φ ∈/ conv(Aut(H3)). Assuming the contrary, suppose Φis a convex combination of 3 Φk ∈ Aut(H ), k =1, 2, ..., N. As observed in Example 4, all these doubly stochastic M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61 59 maps on H3 can be extended to doubly stochastic maps on M3. We see that the extension Φgiven by

∗ ∗ ∗ ∈M3 Φ(X):=V1 XV1 + V2 XV2 + V3 XV3 (X ), is a convex combination of Φk, k =1, 2, ..., N. However, it has been noted in [15], based on Choi’s characterization of extreme completely positive doubly stochastic maps, that 3 Φis an extreme point of the set of all doubly stochastic maps on M . Thus, Φk = Φfor 3 all k. It follows that Φ =Φk ∈ Aut(H ). However, considering the matrix Ei (i =1, 2) 3 in H with one in the (i, i)slot and zeros elsewhere, we see that E1 ◦ E2 = 0 while Φ(E1) ◦Φ(E2) =0. Thus, Φ cannot be an algebra automorphism, yielding a contradiction. Hence, we have (20).

8. Extreme points of DS(V )

As DS(V )is compact and convex, by the well-known Minkowski–Carathéodory The- orem, DS(V )is the convex hull of (the set of) its extreme points. While the description of all extreme points of DS(V ) seems to be a diﬃcult task, in the result below, we show that when V is simple, Jordan algebra automorphisms are extreme points of DS(V ). We ﬁrst prove a simple proposition.

Proposition 2. Let V be a simple Euclidean Jordan algebra and Φ ∈ DS(V ). Then,

||Φ(x)|| ≤ ||x|| (x ∈ V ).

Proof. Since V is simple, the given inner product on V is a positive multiple of the canonical inner product; we may assume without loss of generality that V carries the canonical inner product and ||x||2 = ||x|| for all x. Now, let x ∈ V and y := Φ(x). As V is simple, the implication (b) ⇒ (a)in Theorem 6 shows that y =Ψ(x), where Ψ is a convex combination of algebra automorphisms of V . Now, algebra automorphisms preserve the 2-norm of a vector (cf. Theorem 7), we see (by triangle inequality) that

||Ψ(x)||2 ≤||x||2. Thus, ||Φ(x)|| ≤||x||. 2

In the result below, we let Ext(DS(V )) denote the set of all extreme points of DS(V ).

Theorem 10. Suppose V is simple. Then

Aut(V ) ⊆ Ext(DS(V )).

Moreover, when n ≥ 3,

Aut(Ln) = Ext(DS(Ln)). 60 M.S. Gowda / Linear Algebra and its Applications 528 (2017) 40–61

|| || || || Proof. We assume that V carries the canonical inner product, so x = x2 for all ∈ ∈ N x V . Let Φ Aut(V )and suppose it is written as a convex combination Φ = 1 λk Φk of doubly stochastic maps on V . As Jordan algebra automorphisms preserve the 2-norm (thus, in our setting, the inner product norm), we have, by the above proposition,

N N ||x|| = ||Φ(x)|| ≤ λk ||Φk(x)|| ≤ λk ||x|| = ||x|| 1 1 for all x ∈ V . As the inner product norm is strictly convex, it follows that for each x ∈ V , and k, Φk(x)is a positive multiple of Φ(x). Fix k and write Φk(x) = β(x) Φ(x), where β(x) > 0. As Φand Φk preserve the trace, we see that β(x) = 1 whenever tr(x) =0. Thus, Φk(x) =Φ(x) whenever tr(x) =0. As the set of all such vectors form a dense set in V , by continuity, we see that Φk(x) =Φ(x)for all x ∈ V . Hence Φk =Φ. As this holds for every k, we see that Φis an extreme point of DS(V ). Now the statement about Ln follows from Theorem 9. 2

Concluding Remarks. In this paper, we studied some properties of positive and doubly stochastic maps, and their connection to majorization. A number of questions remain, some of which are listed below.

• In Theorem 6, does the implication (c) ⇒ (b)hold for a general V ? The resolution of this question depends on whether any two Jordan frames can be mapped to each other by doubly stochastic maps (on a general algebra). • When do we have the equality conv(Aut(V )) =DS(V )? • What are the extreme points of DS(V )?

Acknowledgements

Thanks are due to the Referees for their comments and suggestions.

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