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C⁎-convexity of norm unit balls
Accepted Manuscript C ∗ -convexity of norm unit balls
Mohsen Kian
PII: DOI: Reference:
S0022-247X(16)30060-9 http://dx.doi.org/10.1016/j.jmaa.2016.04.022 YJMAA 20361
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
16 December 2015
Please cite this article in press as: M. Kian, C ∗ -convexity of norm unit balls, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.04.022
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C ∗ -CONVEXITY OF NORM UNIT BALLS MOHSEN KIAN This paper is dedicated to Professor Richard Martin Aron Abstract. We characterize the norms on B(H) whose unit balls are C ∗ -convex. We call such norms M -norms and investigate their dual norms, named L-norms. We show that the class of L-norms greater than a given norm enjoys a minimum element and the class of M -norms less than a given norm enjoys a maximum element. These minimum and maximum elements will be determined in some cases. Finally, we give a constructive result to obtain M -norms and L-norms on B(H).
1. Introduction Throughout this paper B(H) is the C ∗ -algebra of all bounded linear operators on a Hilbert space H and I denotes the identity operator on H. If dim H = n, we identify B(H) with Mn , the C ∗ -algebra of all n × n matrices with complex entries. We mean by L1 (H) the ∗-algebra of all trace class operators on H. It is well-known that B(H) is identified with L1 (H)∗ , the dual space of L1 (H), under the mapping S → Tr(· S). A subset K of B(H) is called C ∗ -convex if A1 , . . . , Ak ∈ K and C1 , . . . , Ck ∈ B(H) with ki=1 Ci∗ Ci = I implies that ki=1 Ci∗ Ai Ci ∈ K . This kind of convexity has been introduced by Loebl and Paulsen [9] as a non-commutative generalization of the usual linear convexity. For example the sets {T ∈ B(H); 0 ≤ T ≤ I} and {T ∈ B(H); T ≤ M } are C ∗ -convex. It is evident that the C ∗ -convexity of a set K in B(H) implies its convexity in the usual sense. The converse is not however true in general. Various examples and some basic properties of C ∗ -convex sets are presented in [9]. In recent decades, many operator algebraists paid their attention to extend various concepts in the commutative setting to non-commutative cases. Regarding these works, the notion of C ∗ -convexity in C ∗ -algebras has been established as a non-commutative generalization of the linear convexity in linear spaces. Some essential results of convexity theory have been generalized in [3] to C ∗ -convexity. In 2010 Mathematics Subject Classification. 47A30, 15A60, 47A05. Key words and phrases. C ∗ -convex set, unit ball, dual norm . 1
2
M. KIAN
particular, a version of the so-called Hahn–Banach theorem is presented. The operator extension of extreme points, the C ∗ -extreme points have also been introduced and studied, see [4, 5, 7, 9]. Moreover, Magajna [11] established the notion of C ∗ convexity for operator modules and proved some useful separation theorems. We refer the reader to [8, 10, 12, 13] for further results concerning C ∗ -convexity. The main aim of the present paper is to characterize the norms whose unit balls are C ∗ -convex. In Section 2, we introduce M -norms and L-norms and give their properties and investigate some connections between them. In Section 3, we show that the class of L-norms which are greater than an arbitrary norm · , enjoys a minimum element and the class of M -norms which are less than an arbitrary norm · , possesses a minimum element. We determine this minimum and maximum elements in some cases. For example, we will show that the trace norm ·1 is the minimum element in the class of L-norms greater than the operator norm · ∞ . In Section 4, we give a constructive result to obtain M -norms. 2. M -type and L-type Norms We begin this section with the definitions of M -norm and L-norm. Definition 2.1. A norm · on B(H) is an M-norm (is of M -type) if k k Ci∗ Xi Ci ≤ max Xi Xi ∈ B(H), Ci∗ Ci = I . 1≤i≤k i=1
(1)
i=1
The motivation of the next definition will be revealed soon. Definition 2.2. A norm · on B(H) is an L-norm (is of L-type) if k k Ci XCi∗ ≤ X X ∈ B(H), Ci∗ Ci = I . i=1
(2)
i=1
The next lemma is easy to be proved. So we omit its proof. Lemma 2.3. Let · be a norm on B(H). The unit ball of · is C ∗ -convex if and only if · is a M -norm. It is well-known that {T ∈ B(H); T ≤ 1} is C ∗ -convex. So, the operator norm is a M -norm. The dual space of Mn is identified with Mn itself under the duality coupling Y |X = Tr (Y ∗ X) ,
X, Y ∈ Mn .
C ∗ -CONVEXITY OF NORM UNIT BALLS
3
Here, it should be noticed that Y |Z ∗ XZ = ZY Z ∗ |X
(X, Y, Z ∈ Mn ).
(3)
The dual norm · ∗ of a norm · is defined by Y ∗ = sup {|Y |X |;
X ≤ 1} ,
for all Y ∈ Mn .
(4)
A connection between M -norms and L-norms can be stated as follows. Theorem 2.4. A norm · on Mn is a M -norm (resp. L-norm) if and only if its dual norm · ∗ is a L-norm (resp. M -norm). Proof. Suppose that · is a M -norm. Take B ∈ Mn . Let Ci ∈ Mn (i = 1, . . . , k) with ki=1 Ci∗ Ci = I. For every i = 1, . . . , k, it follows from (4) that there exists Ai ∈ Mn with Ai = 1 and Ci BCi∗ ∗ = Ci BCi∗ | Ai = B | Ci∗ Ai Ci . Since · is of M -type, ki=1 Ci∗ Ai Ci ≤ maxi Ai = 1, so that k i=1
Ci BCi∗ ∗
=
k
B |
Ci∗ Ai Ci
i=1
=
k
Tr (B ∗ (Ci∗ Ai Ci ))
i=1
= Tr B ∗
k
Ci∗ Ai Ci
i=1
k Ci∗ Ai Ci ≤ B∗ . = B i=1
Therefore, · ∗ is a L-norm. Suppose conversely that the dual norm ·∗ is a L-norm. If Ai ∈ Mn (i = 1, . . . , k) and ki=1 Ci∗ Ci = I, then there exits B ∈ Mn such that k k B∗ = 1 and B Ci∗ Ai Ci = Ci∗ Ai Ci . i=1
i=1
4
M. KIAN
We have
k k Ci∗ Ai Ci = B Ci∗ Ai Ci i=1
i=1
=
k
Ci BCi∗ |Ai
(by (3))
i=1
≤
k
Ai · Ci BCi∗ ∗
i=1
≤ max Aj
k
1≤j≤k
Ci BCi∗ ∗
i=1
≤ max Aj · B∗ 1≤j≤k
= max Aj , 1≤j≤k
where the last inequality follows from the fact that · ∗ is a L-norm.
Corollary 2.5. Let · ∗ be the dual norm of a norm · on Mn . The unit ball of · ∗ is C ∗ -convex if and only if · is a L-norm. We can summarize the above discussion as follows. If · ∗ is the dual norm of a norm · on Mn and B1 and B1∗ are the unit ball of · and · ∗ , respectively, then B1 is C ∗ -convex ⇐⇒ · is a M -norm ⇐⇒ · ∗ is a L-norm and B1∗ is C ∗ -convex ⇐⇒ · is a L-norm ⇐⇒ · ∗ is a M -norm. Example 2.6. The trace norm · 1 on Mn defined by A1 = Tr(|A|) is a L-norm. Since · 1 is the dual norm of the operator norm, Lemma 2.4 implies that · 1 is a L-norm. Let · be a norm on Mn . There are M -norms ||| · |||(M 1) and ||| · |||(M 2) and L-norms ||| · |||(L1) and ||| · |||(L2) such that ||| · |||(M 1) ≤ · ≤ ||| · |||(M 2) and ||| · |||(L1) ≤ · ≤ ||| · |||(L2) . Since all norms on Mn are equivalent, there are α, β, μ, ν > 0 such that α · ∞ ≤ · ≤ β · ∞ and μ · 1 ≤ · ≤ ν · 1 . It is enough to define ||| · |||(M 1) := α · ∞ , ||| · |||(M 2) := β · ∞ , ||| · |||(L1) := μ · 1 and ||| · |||(L2) := ν · 1 . Recall that a norm · on B(H) is said to be unitarily invariant if U AV = A for all A ∈ B(H) and all unitaries U, V ∈ B(H), while it is called weakly unitarily invariant if U ∗ AU = A for every A ∈ B(H) and every unitary U ∈ B(H). It is
C ∗ -CONVEXITY OF NORM UNIT BALLS
5
easy to see that all M -norms and L-norms on B(H) are weakly unitarily invariant. Moreover, let E be a projection in B(H). If · is a M -norm, then it follows from (1) with C1 = E, C2 = I − E, A1 = A and A2 = 0 that EAE = C1∗ A1 C1 + C2∗ A2 C2 ≤ max {A, 0} = A and if · is a L-norm, then (2) implies EAE ≤ C1 AC1∗ + C2 AC2∗ ≤ A. We reach the following useful lemma. Lemma 2.7. If · is a L-norm or is a M -norm, then EAE ≤ A for every projection E and every A ∈ B(H). Proposition 2.8. Let · be a norm on Mn . (1) If · is a M -norm, then there exists α > 0 such that A = αA∞ for every normal element A ∈ Mn . (2) If · is a L-norm, then there exists α > 0 such that A = αA1 for every normal element A ∈ Mn . Proof. Note that if · is a M -norm or is a L-norm, then it is weakly unitarily invariant. Moreover, every two rank-one orthoprojections P and Q in Mn are unitarily equivalent, i.e., there exists a unitary U such that P = U ∗ QU . It follows that P = U ∗ QU = Q. Now, if A ∈ Mn is normal, then by the spectral theorem, there are orthoprojections Ei and λi ∈ C (i = 1, . . . , n) such that A=
n
λi Ei ;
Ej Ek = δj,k Ek ,
i=1
n
Ei = I.
(5)
i=1
Note that Ei = Ej for every i, j ∈ {1, . . . , n}. We set α := Ei . (1) If · is a M -norm, then αA∞ = α · max |λi | = max λi Ei = max Ei AEi ≤ A, 1≤i≤n
1≤i≤n
1≤i≤n
where the last inequality follows from Lemma 2.7. In addition, since · is a M -norm, we have n n λi Ei = Ei (λi Ei )Ei ≤ max λi Ei = αA∞ . A = 1≤i≤n i=1
i=1
This completes the proof of (1). The proof of (2) is similar to that of (1). Corollary 2.9. Let · be a unitarily invariant norm on Mn .
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M. KIAN
(1) If · is a M -norm, then · = α · ∞ for some α > 0. (2) If · is a L-norm, then · = α · 1 for some α > 0. Proof. If · is unitarily invariant, then for every A ∈ Mn we have A = |A| by the polar decomposition. Proposition 2.8 can be then applied. It follows from the proof of Proposition 2.8 that if a norm · on Mn is weakly unitarily invariant, then E1 = E2 for each rank-one orthoprojections E1 and E2 . If · is a weakly unitarily invariant norm, then we set χ· := E, where E is any rank-one orthoprojection. The numerical radius norm ω(·) is defined on B(H) by ω(T ) = sup{|T x, x |; x ∈ H, x = 1}. Let ω∗ (·) be the dual norm of the numerical radius norm. Proposition 2.10. Let · be a weakly unitarily invariant norm with χ· = 1. (1) If · is a M -norm, then ω(·) ≤ · ≤ · ∞ . (2) If · is a L-norm, then · 1 ≤ · ≤ ω∗ (·). Proof. (1) First take A with A = 1. For any unit vector x = 1, put E := xx∗ . Since E = 1, and EAE = x, Ax E, we have |x, Ax | = EAE = EAE + (I − E)0(I − E) ≤ A = 1
(since · is a M -norm)
so that ω(A) ≤ 1 = A. Next, take B with B∞ = 1. It is known that B can be written as a convex combination of two unitaries, i.e., B = λU + (1 − λ)V for some λ ∈ [0, 1] and two unitaries U and V . Clearly, every unitary matrix has norm 1 for any M -norm · with χ· = 1. Therefore B ≤ λU + (1 − λ)V = 1 = B∞ . (2) If · is a L-norm, then · ∗ is a M -norm. Moreover, χ·∗ = 1. Part (1) now implies that ω(·) ≤ · ∗ ≤ · ∞ . Finally, duality of norms reveals that · 1 ≤ · ≤ ω∗ (·). 3. Minimum norms of L-type and maximum norms of M -type In this section we are going to show that the class of L-norms enjoys a minimum element. Theorem 3.1. Let · be a given norm. If the class of L-norms on B(H) greater than · is not empty, then it possesses a minimum element.
C ∗ -CONVEXITY OF NORM UNIT BALLS
Proof. Define the norm · (u) by
A(u) := sup
k
Ci ACi∗ ;
k
i=1
7
Ci∗ Ci = I
.
(6)
i=1
It is clear that · (u) is greater than · . Let ||| · ||| be a L-norm and greater than · . If A ∈ B(H) and ki=1 Ci∗ Ci = I, then k
Ci ACi∗
≤
i=1
k
|||Ci ACi∗ ||| ≤ |||A|||.
i=1
This ensures that the sup in (6) is finite and A(u) ≤ |||A|||. It is enough to show ∗ that · (u) is a L-norm. For this end, let A ∈ B(H) and m j=1 Dj Dj = I. By definition (6), for every 0 < < 1 and for every j = 1, . . . , m, there exist Cij such that k
Cij∗ Cij
=I
and
(1 −
)Dj ADj∗ (u)
≤
i=1
k
Cij Dj ADj∗ Cij∗ ,
i=1
so that (1 − )
m
Dj ADj∗ (u)
≤
j=1
Dj∗ Cij∗ Cij Dj =
j=1 i=1
Cij Dj ADj∗ Cij∗ .
(7)
j=1 i=1
Since k m
k m
m j=1
Dj∗
k
Cij∗ Cij
Dj =
i=1
m
Dj∗ Dj = I,
j=1
it follows from (6) that k m
Cij Dj ADj∗ Cij∗ ≤ A(u) .
(8)
j=1 i=1
∗ Hence (1 − ) m j=1 Dj ADj (u) ≤ A(u) by (7) and (8). Letting → 0 we conclude that · (u) is a L-norm. Corollary 3.2. Let · be a given norm. If the class of M -norms less than · is not empty, then it possesses a minimum element. Proof. Set P = {||| · |||; ||| · ||| is a M -norm and ||| · ||| ≤ · } and P∗ = {||| · |||∗ ; ||| · ||| ∈ P} = {||| · |||∗ ; ||| · |||∗ is a L-norm and ||| · |||∗ ≥ · ∗ } .
8
M. KIAN
From Theorem 3.1 we deduce that P∗ has a minimum element, say · ∗(u) . Duality of norms and Lemma 2.4 ensure that · (u) is a M -norm. Moreover, · (u) is the maximum element of P by definition. Consider the operator norm on Mn . Theorem 3.1 guarantees that the class of L-norms grater than the operator norm has a minimum element. In the next result we show that the trace norm · 1 is the minimum one along the class of L-norms bounded below by the operator norm. Theorem 3.3. The trace norm ·1 is the minimum element in the class of L-norms greater than the operator norm · ∞ , i.e.,
k k Ci ACi∗ ∞ ; Ci∗ Ci = I . A1 = sup i=1
i=1
Proof. For any matrix A ∈ Mn and every orthonormal system xj (j = 1, . . . , n) we can write n n n |xj , Axj | = Ej AEj ∞ = Ej AEj 1 ≤ A1 , j=1
j=1
j=1
in which Ej is the rank-one orthoprojection to Cxj . Moreover, assume that A = U |A| be the polar decomposition of A with unitary U . Then A1 = Tr(|A|) = Tr(U |A|U ∗ ) = Tr(AU ∗ ).
(9)
Let n
∗
U =
eiθj xj x∗j
(10)
j=1
be the spectral decomposition of unitary U ∗ , where xj ’s are the (column) unit eigenvectors ( so that x∗j ’s are row vectors and xj x∗j = Ej is the rank-one projection onto the subspace Cxj ). Now, it follows from (9) and (10) that A1 = Tr(AU ∗ ) =
n
eiθj Tr(AEj ) =
n
j=1
eiθj xj , Axj ≤
j=1
n
|xj , Axj | ≤
j=1
n
Ej AEj ∞ .
j=1
(11)
This gives that
A1 ≤ sup
k
Ci ACi∗ ∞ ;
i=1
≤ sup
k i=1
≤ A1
k
Ci∗ Ci = I
i=1
Ci ACi∗ 1 ;
k
Ci∗ Ci = I
i=1
(Since · 1 is a L-norm ),
C ∗ -CONVEXITY OF NORM UNIT BALLS
9
which implies the desired result. We give another example for Theorem 3.1 in the next theorem.
Theorem 3.4. The trace norm ·1 is the minimum element in the class of L-norms greater than the numerical radius norm ω(·). Proof. Assume that · is a L-norm with ω(·) ≤ · . It follows from the proof of Theorem 3.3 that for every A ∈ Mn , there exits an orthonormal system of vectors {xj } such that A1 =
n
|xj , Axj | ≤
j=1
n
ω(Ej AEj )
j=1
≤
n
Ej AEj ≤ A,
j=1
where Ej = xj x∗j . This ensures that · 1 is the minimum L-norm greater than the numerical radius norm. Corollary 3.5. The operator norm · ∞ is the maximum element in the class of M -norms which are smaller than ω∗ (·), the dual norm of ω(·). Proof. Apply Corollary 3.2 and Theorem 3.4.
A natural question is that does the class of L-norms ||| · ||| which are less than a norm · contain a maximum element? or does the class of M -norms ||| · ||| which are greater than a norm · contain a minimum element? We answer these questions in some special cases below. First we need the following simple lemma. We omit its proof. Lemma 3.6. If P and Q are orthoprojections in B(H), then P XP + QY Q1 = P XP 1 + QY Q1 for every X, Y ∈ L1 (H). Theorem 3.7. The norm n · ω(·) is the minimum element in the class of M -norms greater than the trace norm · 1 . Proof. Note first that the unit ball of Mn with respect to the numerical radius norm ω(·) is C ∗ -convex [9] and so ω(·) is a M -norm. For any A ∈ Mn there exists an orthonormal system of vectors {xj }nj=1 of Cn such that A1 = nj=1 |xj , Axj | ≤ n j=1 ω(A) = n ω(A). So, n ω(·) is a M -norm greater than the trace norm · 1 . Now let · be a M -norm greater than the trace norm. For any A ∈ Mn , there is x ∈ Cn such that x = 1 and ω(A) = |x, Ax |. Starting with x1 := x, construct an
10
M. KIAN
orthonormal system {xj } in Cn . Assume that Uj is the unitary such that Uj xj = x. Let Ej := xj x∗j be the rank-one orthoprojection to Cxj . Note that Ej Uj∗ AUj Ej 1 = xj , (Uj∗ AUj )xj Ej 1 = |xj , (Uj∗ AUj )xj | = |Uj xj , AUj xj | = |x, Ax | = ω(A).
(12)
Since · is weakly unitary invariant and · 1 ≤ · , we have n · ω(A) = n |x, Ax | =
n
Ej Uj∗ AUj Ej 1
j=1
n ∗ = Ej Uj AUj Ej (by Lemma 3.6) j=1 1 n Ej Uj∗ AUj Ej ≤ max Uj∗ AUj = A. ≤ 1≤j≤n j=1
This gives that n · ω(·) is the minimum element.
Corollary 3.8. The norm ω∗n(·) is the maximum element in the class of L-norms smaller than the operator norm · ∞ . Proof. This follows from Theorem 3.7 via the duality of corresponding norms.
4. Construction of M -norms and L-norms If f : (0, ∞) → (0, ∞) is an operator concave function, the Jensen’s operator inequality [6] implies that k k f Cj∗ Aj Cj ≥ Cj∗ f (Aj )Cj (13) j=1
for all Aj ≥ 0 and Cj ∈ B(H) with
k
j=1
j=1
Cj∗ Cj = I.
Theorem 4.1. Assume that ϕ(t), ψ(t) > 0 are matrix concave functions on (0, 1). The set
ϕ(A) X Uϕ,ψ = X; ≥ 0, for some 0 ≤ A ≤ I X ∗ ψ(A) is C ∗ -convex. k
Ci∗ Ci = I. If X1 , . . . , Xk ∈ Uϕ,ψ , then
there exist Xi ϕ(Ai ) A1 , . . . , An with 0 ≤ Ai ≤ I (i = 1, . . . , k) such that ≥ 0, for all ∗ Xi ψ(Ai ) i = 1, . . . , k. It follows that Proof. Let Ci ∈ B(H) with
i=1
ϕ(Ai ) ≥ Xi ψ(Ai )−1 Xi∗ ,
i = 1, . . . , k.
(14)
C ∗ -CONVEXITY OF NORM UNIT BALLS
Therefore ϕ
k
Ci∗ Ai Ci
≥
i=1
k
Ci∗ ϕ(Ai )Ci
11
(by matrix concavity of ϕ)
i=1
≥
k
Ci∗ Xi ψ(Ai )−1 Xi∗ Ci
(by (14)).
(15)
i=1
It is easy to verify that (see for example [1, 2]) the function g(X, Y ) = Xψ(Y )−1 X ∗ : B(H) × B(H)+ → B(H)+ is jointly convex. Hence k k −1 k k k Ci∗ Xi Ci ψ Ci∗ Ai Ci Ci∗ Xi∗ Ci = g Ci∗ Xi Ci , Ci∗ Ai Ci i=1
i=1
i=1
i=1
≤
k
i=1
Ci∗ g(Xi , Ai )Ci
i=1
=
k
Ci∗ Xi ψ(Ai )−1 Xi∗ Ci . (16)
i=1
Combining (15) and (16) we get k k k −1 k Ci∗ Ai Ci ≥ Ci∗ Xi Ci ψ Ci∗ Ai Ci Ci∗ Xi∗ Ci . ϕ i=1
It follows that
i=1
i=1
i=1
k Ci∗ Xi∗ Ci ϕ( ki=1 Ci∗ Ai Ci ) i=1 ∗ ≥ 0. k ∗ C X C ψ( ki=1 Ci∗ Ai Ci ) i i i=1 i Noting that 0 ≤ ki=1 Ci∗ Ai Ci ≤ ki=1 Ci∗ Ci = I, we conclude that ki=1 Ci∗ Xi Ci ∈ Uϕ,ψ .
Theorem 4.1 provides a way of constructing M -norms. If we find any norm such that its unit ball coincides with Uϕ,ψ for some proper matrix concave functions ϕ and ψ, then that norm would be a M -norm by Lemma 2.3. An obvious example is the spectral norm · ∞ , where the unit ball coincides with Uϕ,ψ with ϕ(t) = ψ(t) = 1, since
I A ≥ 0. A∞ ≤ 1 ⇐⇒ A∗ I As another example, consider the numerical radius norm ω(·). It is known that [1] that for X ∈ Mn , ω(X) ≤ 1 if and only if there exists a Hermitian matrix H I +H X with −I ≤ H ≤ I such that is positive. ∗ X I −H
12
M. KIAN
With A =
I+H 2
this yields that 0 ≤ A ≤ I and
2A X ≥ 0, for some 0 ≤ A ≤ I. ω(X) ≤ 1 ⇐⇒ X ∗ 2(I − A)
(17)
If ϕ(t) = 2t and ψ(t) = 2(1 − t), then ϕ and ψ are matrix concave functions on (0, 1). Theorem 4.1 then ensures the C ∗ -convexity of U2t,2(1−t) . In addition, U2t,2(1−t) = {X; ω(X) ≤ 1} by (17). This implies that the numerical radius norm ω(·) is a M -norm. Acknowledgement. This Research was in part supported by a grant from IPM (No.94470044). References [1] R. Bhatia, Positive Definite Matrices. Princeton University Press, Princeton, 2007. [2] E. Effros and F. Hansen, Non-commutative perspectives, Ann. Funct. Anal. 5, (2014), 74–79. [3] E.G. Effros and S. Winkler, Matrix Convexity: Operator analogues of the Bipolar and Hahn– Banach Theorems, J. Funct. Anal. 144 (1997), 117–152. [4] D. R. Farenick, C ∗ -extreme points of some compact C ∗ -convex sets, Proc. Amer. Math. Soc. 118 (1993), 765–775. [5] D.R. Farenick and P.B. Morenz, C ∗ -extereme points in the generalised state space of a C ∗ algebra, Trans. Amer. Math. Soc. 349 (1997), 1725–1748. [6] F. Hansen and G.K. Pedersen, Jensen’s operator inequality, Bull. London Math. Soc. 35 (2003), no. 4, 553–564. [7] A. Hopenwasser, R. L. Moore, V. I. Paulsen, C ∗ -extereme points, Trans. Amer. Math. Soc. 163 (1981), 291–307. [8] M. Kian, Epigraph of operator functions, arXiv:1512.05529, to apear in Quaestiones Mathematicae. [9] R. I. Loebl and V. I. Paulsen, Some remarks on C ∗ -convexity, Linear Algebra Appl. 35 (1981), 63–78. [10] P. B. Morenz, The structure of C ∗ -convex sets, Canad. Math. J. Math. 46 (1994), 1007–1026. [11] B. Magajna, C ∗ -convex sets and completely bounded bimodule homomorphisms, Proc. Roy. Soc. Endinburgh Sect. A. 130 (2000), 375-387. [12] B. Magajna, C ∗ -convexity and the numerical range, Canad. Math. Bull., 43 (2000), 193–207. [13] C. Webster and S. Winkler, The Krein–Milman Theorem in operator convexity, Trans. Amer. Math. Soc. 351 (1999), 307–322. Mohsen Kian: Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran. E-mail address: [email protected] and [email protected]
Mohsen Kian
PII: DOI: Reference:
S0022-247X(16)30060-9 http://dx.doi.org/10.1016/j.jmaa.2016.04.022 YJMAA 20361
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
16 December 2015
Please cite this article in press as: M. Kian, C ∗ -convexity of norm unit balls, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.04.022
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C ∗ -CONVEXITY OF NORM UNIT BALLS MOHSEN KIAN This paper is dedicated to Professor Richard Martin Aron Abstract. We characterize the norms on B(H) whose unit balls are C ∗ -convex. We call such norms M -norms and investigate their dual norms, named L-norms. We show that the class of L-norms greater than a given norm enjoys a minimum element and the class of M -norms less than a given norm enjoys a maximum element. These minimum and maximum elements will be determined in some cases. Finally, we give a constructive result to obtain M -norms and L-norms on B(H).
1. Introduction Throughout this paper B(H) is the C ∗ -algebra of all bounded linear operators on a Hilbert space H and I denotes the identity operator on H. If dim H = n, we identify B(H) with Mn , the C ∗ -algebra of all n × n matrices with complex entries. We mean by L1 (H) the ∗-algebra of all trace class operators on H. It is well-known that B(H) is identified with L1 (H)∗ , the dual space of L1 (H), under the mapping S → Tr(· S). A subset K of B(H) is called C ∗ -convex if A1 , . . . , Ak ∈ K and C1 , . . . , Ck ∈ B(H) with ki=1 Ci∗ Ci = I implies that ki=1 Ci∗ Ai Ci ∈ K . This kind of convexity has been introduced by Loebl and Paulsen [9] as a non-commutative generalization of the usual linear convexity. For example the sets {T ∈ B(H); 0 ≤ T ≤ I} and {T ∈ B(H); T ≤ M } are C ∗ -convex. It is evident that the C ∗ -convexity of a set K in B(H) implies its convexity in the usual sense. The converse is not however true in general. Various examples and some basic properties of C ∗ -convex sets are presented in [9]. In recent decades, many operator algebraists paid their attention to extend various concepts in the commutative setting to non-commutative cases. Regarding these works, the notion of C ∗ -convexity in C ∗ -algebras has been established as a non-commutative generalization of the linear convexity in linear spaces. Some essential results of convexity theory have been generalized in [3] to C ∗ -convexity. In 2010 Mathematics Subject Classification. 47A30, 15A60, 47A05. Key words and phrases. C ∗ -convex set, unit ball, dual norm . 1
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M. KIAN
particular, a version of the so-called Hahn–Banach theorem is presented. The operator extension of extreme points, the C ∗ -extreme points have also been introduced and studied, see [4, 5, 7, 9]. Moreover, Magajna [11] established the notion of C ∗ convexity for operator modules and proved some useful separation theorems. We refer the reader to [8, 10, 12, 13] for further results concerning C ∗ -convexity. The main aim of the present paper is to characterize the norms whose unit balls are C ∗ -convex. In Section 2, we introduce M -norms and L-norms and give their properties and investigate some connections between them. In Section 3, we show that the class of L-norms which are greater than an arbitrary norm · , enjoys a minimum element and the class of M -norms which are less than an arbitrary norm · , possesses a minimum element. We determine this minimum and maximum elements in some cases. For example, we will show that the trace norm ·1 is the minimum element in the class of L-norms greater than the operator norm · ∞ . In Section 4, we give a constructive result to obtain M -norms. 2. M -type and L-type Norms We begin this section with the definitions of M -norm and L-norm. Definition 2.1. A norm · on B(H) is an M-norm (is of M -type) if k k Ci∗ Xi Ci ≤ max Xi Xi ∈ B(H), Ci∗ Ci = I . 1≤i≤k i=1
(1)
i=1
The motivation of the next definition will be revealed soon. Definition 2.2. A norm · on B(H) is an L-norm (is of L-type) if k k Ci XCi∗ ≤ X X ∈ B(H), Ci∗ Ci = I . i=1
(2)
i=1
The next lemma is easy to be proved. So we omit its proof. Lemma 2.3. Let · be a norm on B(H). The unit ball of · is C ∗ -convex if and only if · is a M -norm. It is well-known that {T ∈ B(H); T ≤ 1} is C ∗ -convex. So, the operator norm is a M -norm. The dual space of Mn is identified with Mn itself under the duality coupling Y |X = Tr (Y ∗ X) ,
X, Y ∈ Mn .
C ∗ -CONVEXITY OF NORM UNIT BALLS
3
Here, it should be noticed that Y |Z ∗ XZ = ZY Z ∗ |X
(X, Y, Z ∈ Mn ).
(3)
The dual norm · ∗ of a norm · is defined by Y ∗ = sup {|Y |X |;
X ≤ 1} ,
for all Y ∈ Mn .
(4)
A connection between M -norms and L-norms can be stated as follows. Theorem 2.4. A norm · on Mn is a M -norm (resp. L-norm) if and only if its dual norm · ∗ is a L-norm (resp. M -norm). Proof. Suppose that · is a M -norm. Take B ∈ Mn . Let Ci ∈ Mn (i = 1, . . . , k) with ki=1 Ci∗ Ci = I. For every i = 1, . . . , k, it follows from (4) that there exists Ai ∈ Mn with Ai = 1 and Ci BCi∗ ∗ = Ci BCi∗ | Ai = B | Ci∗ Ai Ci . Since · is of M -type, ki=1 Ci∗ Ai Ci ≤ maxi Ai = 1, so that k i=1
Ci BCi∗ ∗
=
k
B |
Ci∗ Ai Ci
i=1
=
k
Tr (B ∗ (Ci∗ Ai Ci ))
i=1
= Tr B ∗
k
Ci∗ Ai Ci
i=1
k Ci∗ Ai Ci ≤ B∗ . = B i=1
Therefore, · ∗ is a L-norm. Suppose conversely that the dual norm ·∗ is a L-norm. If Ai ∈ Mn (i = 1, . . . , k) and ki=1 Ci∗ Ci = I, then there exits B ∈ Mn such that k k B∗ = 1 and B Ci∗ Ai Ci = Ci∗ Ai Ci . i=1
i=1
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M. KIAN
We have
k k Ci∗ Ai Ci = B Ci∗ Ai Ci i=1
i=1
=
k
Ci BCi∗ |Ai
(by (3))
i=1
≤
k
Ai · Ci BCi∗ ∗
i=1
≤ max Aj
k
1≤j≤k
Ci BCi∗ ∗
i=1
≤ max Aj · B∗ 1≤j≤k
= max Aj , 1≤j≤k
where the last inequality follows from the fact that · ∗ is a L-norm.
Corollary 2.5. Let · ∗ be the dual norm of a norm · on Mn . The unit ball of · ∗ is C ∗ -convex if and only if · is a L-norm. We can summarize the above discussion as follows. If · ∗ is the dual norm of a norm · on Mn and B1 and B1∗ are the unit ball of · and · ∗ , respectively, then B1 is C ∗ -convex ⇐⇒ · is a M -norm ⇐⇒ · ∗ is a L-norm and B1∗ is C ∗ -convex ⇐⇒ · is a L-norm ⇐⇒ · ∗ is a M -norm. Example 2.6. The trace norm · 1 on Mn defined by A1 = Tr(|A|) is a L-norm. Since · 1 is the dual norm of the operator norm, Lemma 2.4 implies that · 1 is a L-norm. Let · be a norm on Mn . There are M -norms ||| · |||(M 1) and ||| · |||(M 2) and L-norms ||| · |||(L1) and ||| · |||(L2) such that ||| · |||(M 1) ≤ · ≤ ||| · |||(M 2) and ||| · |||(L1) ≤ · ≤ ||| · |||(L2) . Since all norms on Mn are equivalent, there are α, β, μ, ν > 0 such that α · ∞ ≤ · ≤ β · ∞ and μ · 1 ≤ · ≤ ν · 1 . It is enough to define ||| · |||(M 1) := α · ∞ , ||| · |||(M 2) := β · ∞ , ||| · |||(L1) := μ · 1 and ||| · |||(L2) := ν · 1 . Recall that a norm · on B(H) is said to be unitarily invariant if U AV = A for all A ∈ B(H) and all unitaries U, V ∈ B(H), while it is called weakly unitarily invariant if U ∗ AU = A for every A ∈ B(H) and every unitary U ∈ B(H). It is
C ∗ -CONVEXITY OF NORM UNIT BALLS
5
easy to see that all M -norms and L-norms on B(H) are weakly unitarily invariant. Moreover, let E be a projection in B(H). If · is a M -norm, then it follows from (1) with C1 = E, C2 = I − E, A1 = A and A2 = 0 that EAE = C1∗ A1 C1 + C2∗ A2 C2 ≤ max {A, 0} = A and if · is a L-norm, then (2) implies EAE ≤ C1 AC1∗ + C2 AC2∗ ≤ A. We reach the following useful lemma. Lemma 2.7. If · is a L-norm or is a M -norm, then EAE ≤ A for every projection E and every A ∈ B(H). Proposition 2.8. Let · be a norm on Mn . (1) If · is a M -norm, then there exists α > 0 such that A = αA∞ for every normal element A ∈ Mn . (2) If · is a L-norm, then there exists α > 0 such that A = αA1 for every normal element A ∈ Mn . Proof. Note that if · is a M -norm or is a L-norm, then it is weakly unitarily invariant. Moreover, every two rank-one orthoprojections P and Q in Mn are unitarily equivalent, i.e., there exists a unitary U such that P = U ∗ QU . It follows that P = U ∗ QU = Q. Now, if A ∈ Mn is normal, then by the spectral theorem, there are orthoprojections Ei and λi ∈ C (i = 1, . . . , n) such that A=
n
λi Ei ;
Ej Ek = δj,k Ek ,
i=1
n
Ei = I.
(5)
i=1
Note that Ei = Ej for every i, j ∈ {1, . . . , n}. We set α := Ei . (1) If · is a M -norm, then αA∞ = α · max |λi | = max λi Ei = max Ei AEi ≤ A, 1≤i≤n
1≤i≤n
1≤i≤n
where the last inequality follows from Lemma 2.7. In addition, since · is a M -norm, we have n n λi Ei = Ei (λi Ei )Ei ≤ max λi Ei = αA∞ . A = 1≤i≤n i=1
i=1
This completes the proof of (1). The proof of (2) is similar to that of (1). Corollary 2.9. Let · be a unitarily invariant norm on Mn .
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(1) If · is a M -norm, then · = α · ∞ for some α > 0. (2) If · is a L-norm, then · = α · 1 for some α > 0. Proof. If · is unitarily invariant, then for every A ∈ Mn we have A = |A| by the polar decomposition. Proposition 2.8 can be then applied. It follows from the proof of Proposition 2.8 that if a norm · on Mn is weakly unitarily invariant, then E1 = E2 for each rank-one orthoprojections E1 and E2 . If · is a weakly unitarily invariant norm, then we set χ· := E, where E is any rank-one orthoprojection. The numerical radius norm ω(·) is defined on B(H) by ω(T ) = sup{|T x, x |; x ∈ H, x = 1}. Let ω∗ (·) be the dual norm of the numerical radius norm. Proposition 2.10. Let · be a weakly unitarily invariant norm with χ· = 1. (1) If · is a M -norm, then ω(·) ≤ · ≤ · ∞ . (2) If · is a L-norm, then · 1 ≤ · ≤ ω∗ (·). Proof. (1) First take A with A = 1. For any unit vector x = 1, put E := xx∗ . Since E = 1, and EAE = x, Ax E, we have |x, Ax | = EAE = EAE + (I − E)0(I − E) ≤ A = 1
(since · is a M -norm)
so that ω(A) ≤ 1 = A. Next, take B with B∞ = 1. It is known that B can be written as a convex combination of two unitaries, i.e., B = λU + (1 − λ)V for some λ ∈ [0, 1] and two unitaries U and V . Clearly, every unitary matrix has norm 1 for any M -norm · with χ· = 1. Therefore B ≤ λU + (1 − λ)V = 1 = B∞ . (2) If · is a L-norm, then · ∗ is a M -norm. Moreover, χ·∗ = 1. Part (1) now implies that ω(·) ≤ · ∗ ≤ · ∞ . Finally, duality of norms reveals that · 1 ≤ · ≤ ω∗ (·). 3. Minimum norms of L-type and maximum norms of M -type In this section we are going to show that the class of L-norms enjoys a minimum element. Theorem 3.1. Let · be a given norm. If the class of L-norms on B(H) greater than · is not empty, then it possesses a minimum element.
C ∗ -CONVEXITY OF NORM UNIT BALLS
Proof. Define the norm · (u) by
A(u) := sup
k
Ci ACi∗ ;
k
i=1
7
Ci∗ Ci = I
.
(6)
i=1
It is clear that · (u) is greater than · . Let ||| · ||| be a L-norm and greater than · . If A ∈ B(H) and ki=1 Ci∗ Ci = I, then k
Ci ACi∗
≤
i=1
k
|||Ci ACi∗ ||| ≤ |||A|||.
i=1
This ensures that the sup in (6) is finite and A(u) ≤ |||A|||. It is enough to show ∗ that · (u) is a L-norm. For this end, let A ∈ B(H) and m j=1 Dj Dj = I. By definition (6), for every 0 < < 1 and for every j = 1, . . . , m, there exist Cij such that k
Cij∗ Cij
=I
and
(1 −
)Dj ADj∗ (u)
≤
i=1
k
Cij Dj ADj∗ Cij∗ ,
i=1
so that (1 − )
m
Dj ADj∗ (u)
≤
j=1
Dj∗ Cij∗ Cij Dj =
j=1 i=1
Cij Dj ADj∗ Cij∗ .
(7)
j=1 i=1
Since k m
k m
m j=1
Dj∗
k
Cij∗ Cij
Dj =
i=1
m
Dj∗ Dj = I,
j=1
it follows from (6) that k m
Cij Dj ADj∗ Cij∗ ≤ A(u) .
(8)
j=1 i=1
∗ Hence (1 − ) m j=1 Dj ADj (u) ≤ A(u) by (7) and (8). Letting → 0 we conclude that · (u) is a L-norm. Corollary 3.2. Let · be a given norm. If the class of M -norms less than · is not empty, then it possesses a minimum element. Proof. Set P = {||| · |||; ||| · ||| is a M -norm and ||| · ||| ≤ · } and P∗ = {||| · |||∗ ; ||| · ||| ∈ P} = {||| · |||∗ ; ||| · |||∗ is a L-norm and ||| · |||∗ ≥ · ∗ } .
8
M. KIAN
From Theorem 3.1 we deduce that P∗ has a minimum element, say · ∗(u) . Duality of norms and Lemma 2.4 ensure that · (u) is a M -norm. Moreover, · (u) is the maximum element of P by definition. Consider the operator norm on Mn . Theorem 3.1 guarantees that the class of L-norms grater than the operator norm has a minimum element. In the next result we show that the trace norm · 1 is the minimum one along the class of L-norms bounded below by the operator norm. Theorem 3.3. The trace norm ·1 is the minimum element in the class of L-norms greater than the operator norm · ∞ , i.e.,
k k Ci ACi∗ ∞ ; Ci∗ Ci = I . A1 = sup i=1
i=1
Proof. For any matrix A ∈ Mn and every orthonormal system xj (j = 1, . . . , n) we can write n n n |xj , Axj | = Ej AEj ∞ = Ej AEj 1 ≤ A1 , j=1
j=1
j=1
in which Ej is the rank-one orthoprojection to Cxj . Moreover, assume that A = U |A| be the polar decomposition of A with unitary U . Then A1 = Tr(|A|) = Tr(U |A|U ∗ ) = Tr(AU ∗ ).
(9)
Let n
∗
U =
eiθj xj x∗j
(10)
j=1
be the spectral decomposition of unitary U ∗ , where xj ’s are the (column) unit eigenvectors ( so that x∗j ’s are row vectors and xj x∗j = Ej is the rank-one projection onto the subspace Cxj ). Now, it follows from (9) and (10) that A1 = Tr(AU ∗ ) =
n
eiθj Tr(AEj ) =
n
j=1
eiθj xj , Axj ≤
j=1
n
|xj , Axj | ≤
j=1
n
Ej AEj ∞ .
j=1
(11)
This gives that
A1 ≤ sup
k
Ci ACi∗ ∞ ;
i=1
≤ sup
k i=1
≤ A1
k
Ci∗ Ci = I
i=1
Ci ACi∗ 1 ;
k
Ci∗ Ci = I
i=1
(Since · 1 is a L-norm ),
C ∗ -CONVEXITY OF NORM UNIT BALLS
9
which implies the desired result. We give another example for Theorem 3.1 in the next theorem.
Theorem 3.4. The trace norm ·1 is the minimum element in the class of L-norms greater than the numerical radius norm ω(·). Proof. Assume that · is a L-norm with ω(·) ≤ · . It follows from the proof of Theorem 3.3 that for every A ∈ Mn , there exits an orthonormal system of vectors {xj } such that A1 =
n
|xj , Axj | ≤
j=1
n
ω(Ej AEj )
j=1
≤
n
Ej AEj ≤ A,
j=1
where Ej = xj x∗j . This ensures that · 1 is the minimum L-norm greater than the numerical radius norm. Corollary 3.5. The operator norm · ∞ is the maximum element in the class of M -norms which are smaller than ω∗ (·), the dual norm of ω(·). Proof. Apply Corollary 3.2 and Theorem 3.4.
A natural question is that does the class of L-norms ||| · ||| which are less than a norm · contain a maximum element? or does the class of M -norms ||| · ||| which are greater than a norm · contain a minimum element? We answer these questions in some special cases below. First we need the following simple lemma. We omit its proof. Lemma 3.6. If P and Q are orthoprojections in B(H), then P XP + QY Q1 = P XP 1 + QY Q1 for every X, Y ∈ L1 (H). Theorem 3.7. The norm n · ω(·) is the minimum element in the class of M -norms greater than the trace norm · 1 . Proof. Note first that the unit ball of Mn with respect to the numerical radius norm ω(·) is C ∗ -convex [9] and so ω(·) is a M -norm. For any A ∈ Mn there exists an orthonormal system of vectors {xj }nj=1 of Cn such that A1 = nj=1 |xj , Axj | ≤ n j=1 ω(A) = n ω(A). So, n ω(·) is a M -norm greater than the trace norm · 1 . Now let · be a M -norm greater than the trace norm. For any A ∈ Mn , there is x ∈ Cn such that x = 1 and ω(A) = |x, Ax |. Starting with x1 := x, construct an
10
M. KIAN
orthonormal system {xj } in Cn . Assume that Uj is the unitary such that Uj xj = x. Let Ej := xj x∗j be the rank-one orthoprojection to Cxj . Note that Ej Uj∗ AUj Ej 1 = xj , (Uj∗ AUj )xj Ej 1 = |xj , (Uj∗ AUj )xj | = |Uj xj , AUj xj | = |x, Ax | = ω(A).
(12)
Since · is weakly unitary invariant and · 1 ≤ · , we have n · ω(A) = n |x, Ax | =
n
Ej Uj∗ AUj Ej 1
j=1
n ∗ = Ej Uj AUj Ej (by Lemma 3.6) j=1 1 n Ej Uj∗ AUj Ej ≤ max Uj∗ AUj = A. ≤ 1≤j≤n j=1
This gives that n · ω(·) is the minimum element.
Corollary 3.8. The norm ω∗n(·) is the maximum element in the class of L-norms smaller than the operator norm · ∞ . Proof. This follows from Theorem 3.7 via the duality of corresponding norms.
4. Construction of M -norms and L-norms If f : (0, ∞) → (0, ∞) is an operator concave function, the Jensen’s operator inequality [6] implies that k k f Cj∗ Aj Cj ≥ Cj∗ f (Aj )Cj (13) j=1
for all Aj ≥ 0 and Cj ∈ B(H) with
k
j=1
j=1
Cj∗ Cj = I.
Theorem 4.1. Assume that ϕ(t), ψ(t) > 0 are matrix concave functions on (0, 1). The set
ϕ(A) X Uϕ,ψ = X; ≥ 0, for some 0 ≤ A ≤ I X ∗ ψ(A) is C ∗ -convex. k
Ci∗ Ci = I. If X1 , . . . , Xk ∈ Uϕ,ψ , then
there exist Xi ϕ(Ai ) A1 , . . . , An with 0 ≤ Ai ≤ I (i = 1, . . . , k) such that ≥ 0, for all ∗ Xi ψ(Ai ) i = 1, . . . , k. It follows that Proof. Let Ci ∈ B(H) with
i=1
ϕ(Ai ) ≥ Xi ψ(Ai )−1 Xi∗ ,
i = 1, . . . , k.
(14)
C ∗ -CONVEXITY OF NORM UNIT BALLS
Therefore ϕ
k
Ci∗ Ai Ci
≥
i=1
k
Ci∗ ϕ(Ai )Ci
11
(by matrix concavity of ϕ)
i=1
≥
k
Ci∗ Xi ψ(Ai )−1 Xi∗ Ci
(by (14)).
(15)
i=1
It is easy to verify that (see for example [1, 2]) the function g(X, Y ) = Xψ(Y )−1 X ∗ : B(H) × B(H)+ → B(H)+ is jointly convex. Hence k k −1 k k k Ci∗ Xi Ci ψ Ci∗ Ai Ci Ci∗ Xi∗ Ci = g Ci∗ Xi Ci , Ci∗ Ai Ci i=1
i=1
i=1
i=1
≤
k
i=1
Ci∗ g(Xi , Ai )Ci
i=1
=
k
Ci∗ Xi ψ(Ai )−1 Xi∗ Ci . (16)
i=1
Combining (15) and (16) we get k k k −1 k Ci∗ Ai Ci ≥ Ci∗ Xi Ci ψ Ci∗ Ai Ci Ci∗ Xi∗ Ci . ϕ i=1
It follows that
i=1
i=1
i=1
k Ci∗ Xi∗ Ci ϕ( ki=1 Ci∗ Ai Ci ) i=1 ∗ ≥ 0. k ∗ C X C ψ( ki=1 Ci∗ Ai Ci ) i i i=1 i Noting that 0 ≤ ki=1 Ci∗ Ai Ci ≤ ki=1 Ci∗ Ci = I, we conclude that ki=1 Ci∗ Xi Ci ∈ Uϕ,ψ .
Theorem 4.1 provides a way of constructing M -norms. If we find any norm such that its unit ball coincides with Uϕ,ψ for some proper matrix concave functions ϕ and ψ, then that norm would be a M -norm by Lemma 2.3. An obvious example is the spectral norm · ∞ , where the unit ball coincides with Uϕ,ψ with ϕ(t) = ψ(t) = 1, since
I A ≥ 0. A∞ ≤ 1 ⇐⇒ A∗ I As another example, consider the numerical radius norm ω(·). It is known that [1] that for X ∈ Mn , ω(X) ≤ 1 if and only if there exists a Hermitian matrix H I +H X with −I ≤ H ≤ I such that is positive. ∗ X I −H
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M. KIAN
With A =
I+H 2
this yields that 0 ≤ A ≤ I and
2A X ≥ 0, for some 0 ≤ A ≤ I. ω(X) ≤ 1 ⇐⇒ X ∗ 2(I − A)
(17)
If ϕ(t) = 2t and ψ(t) = 2(1 − t), then ϕ and ψ are matrix concave functions on (0, 1). Theorem 4.1 then ensures the C ∗ -convexity of U2t,2(1−t) . In addition, U2t,2(1−t) = {X; ω(X) ≤ 1} by (17). This implies that the numerical radius norm ω(·) is a M -norm. Acknowledgement. This Research was in part supported by a grant from IPM (No.94470044). References [1] R. Bhatia, Positive Definite Matrices. Princeton University Press, Princeton, 2007. [2] E. Effros and F. Hansen, Non-commutative perspectives, Ann. Funct. Anal. 5, (2014), 74–79. [3] E.G. Effros and S. Winkler, Matrix Convexity: Operator analogues of the Bipolar and Hahn– Banach Theorems, J. Funct. Anal. 144 (1997), 117–152. [4] D. R. Farenick, C ∗ -extreme points of some compact C ∗ -convex sets, Proc. Amer. Math. Soc. 118 (1993), 765–775. [5] D.R. Farenick and P.B. Morenz, C ∗ -extereme points in the generalised state space of a C ∗ algebra, Trans. Amer. Math. Soc. 349 (1997), 1725–1748. [6] F. Hansen and G.K. Pedersen, Jensen’s operator inequality, Bull. London Math. Soc. 35 (2003), no. 4, 553–564. [7] A. Hopenwasser, R. L. Moore, V. I. Paulsen, C ∗ -extereme points, Trans. Amer. Math. Soc. 163 (1981), 291–307. [8] M. Kian, Epigraph of operator functions, arXiv:1512.05529, to apear in Quaestiones Mathematicae. [9] R. I. Loebl and V. I. Paulsen, Some remarks on C ∗ -convexity, Linear Algebra Appl. 35 (1981), 63–78. [10] P. B. Morenz, The structure of C ∗ -convex sets, Canad. Math. J. Math. 46 (1994), 1007–1026. [11] B. Magajna, C ∗ -convex sets and completely bounded bimodule homomorphisms, Proc. Roy. Soc. Endinburgh Sect. A. 130 (2000), 375-387. [12] B. Magajna, C ∗ -convexity and the numerical range, Canad. Math. Bull., 43 (2000), 193–207. [13] C. Webster and S. Winkler, The Krein–Milman Theorem in operator convexity, Trans. Amer. Math. Soc. 351 (1999), 307–322. Mohsen Kian: Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran. E-mail address: [email protected] and [email protected]