数理解析研究所講究録
第 1996 巻 2016 年 92-98

92

On operational convex combinations
Marie ChodaAbstract

We introduce the notion of operational convex combinations together with operational exetreme points for linear maps ofand show that automorphisms are the operational extreme points of.
the set of the unital completely positive maps on$M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$1

Introduction

In the paper [1], we gave some characterization for unital positive Tr-preserving
from a view point ofconplex matrices
maps of the algebra of, That is, a positive unital ‘lftvon Neumann entropy for states ofpreserves the von Neumann ntropy of a given
preserving map ofstate if and only if plays a role of an automorphism forIn this note, we pick up the set of unital completely positive (called“ ucp”
, we re. That is, for a unital linear map offor short) maps ofplace the property”Tr-preserving and positive”’ to the property”completelypositive”, and investigate that what kind of position the automorphisms
stand in ucp maps.
Here, we shall consider the notion of convexity not only scalar multiplication but also the multiplication via operators and generalize the notion ofconvexity, i.e., we introduce the notion of operational convex conbination.
The motivation for the terminology “‘operational convex conbinations”
comes from the following two definitions: One is Lindblad’s “operationalpartition” in [6] (see [7] or [8]) and the other is Cuntz’s canonical endomorphismin [5]. It seems to be natural for treating the set of ucp maps as.
the set of operational convex conbinations of automorphisms of
We also introduce the notion of operational extreme point, and we show,that automorphisms are operational extreme points in ucp maps of$n\cross n$$M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$$\Phi$$\phi$$M_{n}(\mathbb{C})$$\Phi$$M_{n}(\mathbb{C})$$e’$$\phi.$$M_{n}(\mathbb{C})$$\Phi_{n}$$M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$93
which implies that automorphisms are extreme points in ucp maps ofin the usual sense.

2$M_{n}(\mathbb{C})$Operational partitions

What we need to define a convex sum? In usual we need a probability vector$\lambda=(\lambda_{1}, \cdots, \lambda_{n})$:$\lambda_{i}\geq 0, \sum_{i}\lambda_{i}=1.$

Given a finite subset $x=\{x_{1}, x_{n}\}$ of a vector space $X$ , the vector
is a convex sum of via
Now, we consider such a as a “‘finite partition of 1”’$\lambda^{\rangle}$$\sum_{i}\lambda_{i}x_{i}$$x$$\lambda$

Two generalized notions of finite partition of 1 are given in the fram\’eworkof the non-commutative entropy as follows:
Let be a unital -algebra.
(1) A finite subset $\{x_{1}, x_{k}\}$ ofis called a finite partition of unity$Connes-St\emptyset$byrmer ([4]) if they are nonnegative operators which satisfy that$A$$C^{*}$$A$$\sum_{i=1}^{n}x_{i}=1_{A},$(2) A finite subset $\{x_{1}, x_{k}\}$ of $A$ is called a finite operational partition
in of unity of size by Lindblad ([6]) if
$A$

$k$

$\sum_{i}^{k}x_{i}^{*}x_{i}=1_{A}.$Our main target in this note is a finite subset $\{v_{1}, v_{k}\}$ of non-zeroelements in such thatis a finite operational partition of unity
so that. We call such a set $\{v_{1}, v_{k}\}a$ finite operational
partition of unity of size in , and denote the set of all finite operational
partition of unity in by FOP (A) :$A$$\{v_{1}^{*}, v_{k}^{*}\}$$\sum_{i}^{k}v_{i}v_{i}^{*}=1_{A}$

$k$

$A$

$A$

FOP $(A)= \{\{v_{1}, \cdots, v_{k}\}|0\neq v_{i}\in A, \forall i, \sum_{i}^{k}v_{i}v_{i}^{*}=1_{A}, k=1, 2, \}(2.1)$
We denote by $U(A)$ the set of all unitaries in . Clearly, $U(A)$ is the set ofthe most trivial finite operational partition of unity with the size 1.
$A$

2.1

Unital completely positive (ucp) map$\Phi$

Let
be the-algebra ofmatrices over the complex field
A linear map
on a unital -algebra
is positiveis positive iﬀ$C^{*}$$M_{n}(\mathbb{C})$$\Phi$$\mathbb{C}.$$n\cross n$$C^{*}$$A$$\Phi(a)$94
is positive for all
for all positive $a\in A$ and completely positive iﬀ
definedis the map on
positive integer , where the map$x\inA$.
andfor allby.
We restrict the unital -algebra toIn [3, Theorem 2], Choi gave the following characterization: a linear map
forof
is completely positive iﬀ is of the form
inducing the. Moreover, forby someall
, we may require thatis linearly independentform
so that in the form the number is uniquely determind. Such a form wascalled a ‘canonical’ expression for (see [3, Remark 4
Let us call the uniquely determind number $m$ the size of the$\Phi\otimes 1_{k}$$k$$A\otimes M_{k}(\mathbb{C})$$\Phi\otimes 1_{k}$$y\in M_{k}(\mathbb{C})$$\Phi\otimes 1_{k}(x\otimes y)=\Phi(x)\otimes y$$A$$C^{*}$$\Phi$$M_{n}(\mathbb{C})$$\Phi$$M_{n}(\mathbb{C})$$x\in M_{n}(\mathbb{C})$$\Phi(x)=\sum_{i=1}^{m}v_{i}xv_{i}^{*}$$\{v_{i}\}_{i=1}^{m}$$\{v_{i}\}_{i=1}^{m}\subset M_{n}(\mathbb{C})$$\Phi(x)=\sum_{i=1}^{m}v_{i}xv_{i}^{*}$$\{v_{i}\}_{i}$$m$$\Phi$$\Phi.$

Now we pick up the case where is a unital completely positive (calledused. Then the”’ucp”’ for short) map ofsatisfies thatin the form
inis induced someThis means that each ucp map of.
FOP, the map Adv on
is given by
Given an operator$Adv(x)=vxv^{*},$ $(x\in M)$ . Then the groupof all automorphisms$Aut(M_{n}(\mathbb{C}))=$
Adu $|u\in U(M_{n}(\mathbb{C}))\},$
is written by the formof. Similarly, the set
whereis the group of all unitaries in$UCP(M_{n}(\mathbb{C}))$ of all ucp maps on
is written by the following form:$\Phi$$\{v_{1}, \cdots, v_{m}\}\subseteq M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$$\Phi(x)=\sum_{i=1}^{m}v_{i}xv_{i}^{*},$ $(x\in M_{n}(\mathbb{C}))$$\Phi$$\sum_{i=1}^{m}v_{i}v_{i}^{*}=1.$$\{v_{1},$$M_{n}(\mathbb{C})$$)v_{m}\}$$(M_{n}(\mathbb{C}))$$v\in M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$$Aut(M_{n}(\mathbb{C}))$$M_{n}(\mathbb{C})$$\{$$U(M_{n}(\mathbb{C}))$$M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$$UCP(M_{n}( \mathbb{C}))=\{\sum_{i=1}^{m}Adv_{i}|\{v_{i}\}_{i=1}^{m}\in FOP(M_{n}(\mathbb{C})), m=1, 2, \}(2.2)$

3

Operational Convex Combination

3.1

Operational convexity

andbe a set of linear maps onDefinition 3.1. Let$FOP(M_{n}(\mathbb{C}))$ . We callan operational convex combination of. We also say that a subset
with an operational coeficientsis operational convex if it is closed under allof linear maps on
operational convex combinations.
$M_{n}(\mathbb{C})$$\{\Phi_{i}\}_{i=1}^{m}$$\{v_{i}\}_{i=1}^{m}\in$$\sum_{i=1}^{m}Adv_{i}\circ\Phi_{i}$$\{v_{i}\}_{i=1}^{m}$$\{\Phi_{i}\}_{i=1}^{m}$$S$$M_{n}(\mathbb{C})$We can consider $UCP(M_{n}(\mathbb{C}))$ as the set of all operator convex combi. Moreover $UCP(M_{n}(\mathbb{C}))$ is represented asnations of the group$Aut(M_{n}(\mathbb{C}))$95
the set of all operational convex combinations of the identity $id$ of.
We give some characterization for a role ofin $UCP(M_{n}(\mathbb{C}))$
from a view point of extreme points.
$M_{n}(\mathbb{C})$$Aut(M_{n}(\mathbb{C}))$

3.1.1

Cuntz’s canonical endomorphism as an example

The Cuntz’s canonical endomorphism([3]) is an interesting example inunital completely positive maps of infinite dimensional simple -algebras,
which is given as an operational convex combination of the identity: Let
be isometries on an infinite dimensional Hilbert space $H$
such that. The Cuntz algebrais the -algebra generated by. The map
is given as
for all $x\in O_{n}.$So, in our notation, $\{S_{1\}}8_{2}, \cdots, S_{n}\}\in FOP(O_{n})$ and $\Phi_{n}\in UCP(O_{n})$ .
The left inverse ofplays an inportant role in the theory of Cuntzalgebras and it is given by the form $\Psi(x)=(1/n)\sum_{i}S_{i}^{*}xS_{i},$ $(x\in O_{n})$ .
We remark that is also an operational convex combination of the identity and $\Psi\in UCP(O_{n})$ .
Later we discuss in another paper on the case of unital infinite dimensional-algebras represented by$\Phi_{n}$$C^{*}$$\{S_{1}, S_{2}, \cdots, S_{n}\}$$\sum_{i}S_{i}S_{i}^{*}=1$$\{S_{1}, S_{2}, \cdots, S_{n}\}$$O_{n}$$\Phi_{n}(x)=\sum_{i}S_{i}xS_{i}^{*}$$\Phi_{n}$$\Psi$$C^{*}$$\Phi_{n}$$\Psi$$C^{*}$

3.1.2$O_{n}.$

Operational extreme pointNow let us remember the notion of extreme points. Let be a convex set.
Then a $z\in S$ is an extreme point in if cannot be the convex combination$\lambda x+(1-\lambda)y$ of two points$y\in S$ with $x\neq y$ and $\lambda\in(0,1)$ , i.e., if$z=\lambda x+(1-\lambda)y,$ $(x, y\in S)$ then$x=y=z.$In this note, we say this notion of extreme points an extreme point in the
$S$

$S$

$z$

$x,$

$u\mathcal{S}uat$

sense.

In the usual sense, any automorphism of
can not be expressed
as a convex combination of two automorphisms (see [IO]). However if wereplace a convex combination to an oparational convex combination, then it
is possible as in the following example:$M_{n}(\mathbb{C})$

Example 3.2. Let, and let $\lambda\in(0,1)$ .
be unitaries in$=(1-\lambda)^{1/2}w^{*}$
Let. Then $\{a, b\}\in FOP(M_{n}(\mathbb{C}))$ , andaKld b
the operational convex combination of the automorphisms and ofwith $\Phi=$ Adv and $\Psi=Adw$ with the operational coeﬃcients $\{a, b\}$ is the.
identity map, i.e., $a\Phi(x)a^{*}+b\Psi(x)b^{*}=x$ for all$v,$ $w$$M_{n}(\mathbb{C})$$a=\lambda^{1/2}v^{*}$$\Phi$$x\in M_{n}(\mathbb{C})$$\Psi$$M_{n}(\mathbb{C})$96
Moreover, for a given
with $\Theta=Adu,$ $(u\in U(M_{n}(\mathbb{C}))$ ,if we let
and $b’=(1-\lambda)^{1/2}uw^{*}$ then $\{a’, b’\}\in FOP(M_{n}(\mathbb{C}))$ ,with the operational
and the operational convex conbination of and$\{a’,b’\}$is the automorphism , i.e., $a’\Phi(x)a’*+b’\Psi(x)b’*=\Theta(x)$coeﬃcients.
for all$\Theta\in Aut(M_{n}(\mathbb{C}))$$a’=\lambda^{1/2}uv^{*}$$\Psi$$\Phi$$\Theta$$x\in M_{n}(\mathbb{C})$In the case of operational convex combinations for linear maps andwith an operational coeﬃcient $\{a, b\}\in FOP(M_{n}(\mathbb{C})$ , the mapon
does $(1-\lambda)y$ . Putting this in mind,
corresponds
and the AdbAda(i.e., thelet us define as follows and show that an automorphism ofucp maps with the size 1) is an operational extreme point.
$\Psi$$\Phi$$M_{n}(\mathbb{C})$$0\Phi$$0\Psi$$\lambda x$$M_{n}(\mathbb{C})$Definition 3.3. Let be an operational convex subset of linear maps on. We say that a $\Phi\in S$ is an operational extreme point of if areprensentation of that $\Phi=Ada\circ\Phi_{1}+Adb\circ\Phi_{2},.(\Phi_{i}\in S, (i=1,2),$ $\{a, b\}\in$$FOP(M_{n}(\mathbb{C}))$ implies thatfor someso that $\lambda^{-1}Adao\Phi_{1}=\{1-\lambda\}^{-1}Adbo\Phi_{2}=\Phi.$
$S$

$S$

$M_{n}(\mathbb{C})$$\Phi$$aa^{*}=\lambda 1_{M_{n}(\mathbb{C})},$$bb^{*}=(1-\lambda)1_{M_{n}(\mathbb{C})}$$\lambda\in(0,1)$

isRemark 3.4. (1) If $\{a, b\}\in FOP(M_{n}(\mathbb{C}))$ , then, of course,$Connes-St\emptyset
rmer$’s sense. This defialways a finite partition of unity in the$\Phi\inS$
is represented as annition says that if an operational extreme point$\{a,b\}\in FOP(M_{n}(\mathbb{C}))$ ,$(i=1,2)$ ,operational convex combination ofis a probability vector.
then the Connes Strmer partition$\Phi\inS$(2) If a
is an operational extreme point of a convex subset of, then is an extreme point of in the usual sense.
linear maps on
and is an operationalIn fact, assume thatandextreme point of . Thenso that is an extreme point of in the usual sense.
$\{aa^{*}, bb^{*}\}$$\Phi_{i}\in S,$$\{aa^{*}, bb^{*}\}$

$S$

$S$

$\Phi$$M_{n}(\mathbb{C})$$\Phi=\lambda\Phi_{1}+(1-\lambda)\Phi_{2},$ $(\Phi_{i}\in S)$$S$$\Phi_{1}=\lambda^{-1}\lambda\Phi_{1}=\Phi$$\Phi$$\Phi_{2}=\{1-\lambda\}^{-1}(1-\lambda)\Phi_{2}=\Phi$$S$$\Phi$

The following lemma plays a key role to prove our main theorem:

Lemma 3.5. Assume that $\sum_{i=1}^{m}Adv_{i}=$ Adu for
and $u\in U(M_{n}(\mathbb{C}))$ . Then is a scalar multiple of$v_{i}$$\{v_{i}\}_{i=1}^{m}\in FOP(M_{n}(\mathbb{C}))$$u$

for all $i=1$ , 2,$\cdots,$ $m.$is decomposed into an opTheorem 3.6. If an automorphismof$\Theta=$ Ada $\circ\Phi+Adb\circ\Psi$ via $\{a, b\}\in$erational convex combination that$\Theta$$M_{n}(\mathbb{C})$97
FOP$(M_{7l}(\mathbb{C}))$$M_{n}(\mathbb{C})$and$\Phi,$$u$, then there exist unitaries$v,$ $w\in$and a $\lambda\in(0,1)$ such that$a=\sqrt{\lambda}uv^{*},$

where$\Psi\in UCP(M_{n}(\mathbb{C}))$$b=\sqrt{1-\lambda}uw^{*}$

and$\Phi=A\mathfrak{c}iv,$$\Psi=Adw$(3.1)

is a unitary with $\Theta=Adu.$

As a direct concequence of thls theorem, we have the following:
Corollary 3.7. The automorphism group is a subset of the operational ex.
treme points of the unital completely positive linear maps on$M_{n}(\mathbb{C})$As we remarked in the above, an operational extreme point is an extreme
point in the usual sense, we have the following another consequence:
An automorphism ofis an extreme point of the set of $ucp$ mapsonin the usual sense.
$M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$

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[3] M.D. Choi, Completely positive linear maps on complex matrices, Lin.
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$St\emptyset$

Osaka Kyoiku University
Asahigaoka, Kashiwara 582-8582
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