A㎜EON POSITIVITY IN SYmarRIC IVONOIDS．
Koidhi． KUWANO 〔Re’ceived May 30， 1980〕 0．PRELIMINAR工ES． Asynrnetric monoid M is a sirrple algebraic obj ect whidll satisfies the fo11σvting conditiorls； （a） ryi is a semi−group with unit e． 〔b〕 An involution−like operati㎝★：． M→M is defined and has the fbllow− ing properties； （b．1） （xy〕★＝y「k x★ for xsy∈M。 （b．2〕 xkt＝x for x∈M． There is a c田1㎝ical functor S fr㎝the category of symmetric monoidS to the・ category of symmletric algebras・ Hence va i ous resuユts related to positivity are induced i 1 the c㎝text of s）rimietric monoids． In this note， we shall deal with c（㎎）1ete POsit ivit y in the eategory of sy田metric monoids・ ［0．1］ DEFINITION． Let M be a s）rmaetric monoid． A coMI）1ex−valued fUnction f is said tO be a sy㎜metric character if f satisfies・； （a〕 f（xy）＝f（x）f〔y） forx，y∈M． ’ （b〕f（xft）＝f辰） fbr x・∈M．〔c〕f〔e〕＝1． ・
［0．2］ DEFINITION Let M be a sy㎜etric monoid． Aconlplex−valued ． function f on．M is said to be positively sy頂netric if （・）fb・頗y（λ、，…一，λ。〕・．ψand（・、・ニ…・㌔）・f； 、；jλiちf（x・Xj★）≧° 砲ere n is an a］d）itrary positive integeτ． （b〕 f〔xft〕 ＝f〔x） fOr x∈ M． M“ den・tes the c・as・・f…P・Si・iy・sy・・m・t・i・麺・ti・n…n・M・M’f・ms a ★ − ring wiゼh positive scalar multiplicati《m・41
42
K．KUWANO
Acσnrplex−valued fi瓜cti（m f is called a state if f is positively sylmletric with f（e） ＝1． yOの denotes the convex cone of all states on M． ［0．3］DEFINITICN． Suppose that M and l、 are sy㎜etric血Dnoids． Amapping f：Mr＞Lis called a s）㎜etric h㎝㎝Orphism if
〔a） f（xy） ＝f〔x） f〔y） 〔b〕 f（x★〕 ＝f（x）★ 〔c〕 f〔e）＝el 煽here e and e， denotes． tlle unLits of M and L respectively． We now recall definitions and some basic facts of symmetric algdbras． A synmetric algebra A is an associative algebra with an血voluti㎝σver the co叫）1ex field． Alinear fUnctional f：A−≒〉￠is said to be positive if・f◎c xk）≧O fb・x・A． The c。nv・x…e・f・11 P・Sitiv・㎞・ti。聰1・i・d・n・t・d by A’． Astate is a positive fUncti（mal f with f（1）＝1． 夕〔A〕denotes the set of all states on A． Let A＋denote．the positive cone of A，i．eつ． ㌧・｛・・A；f（・〕≧0（f・め｝．
Suppose that A and B are s）rmmetric algebras． Amappi㎎ψ：． A→B is called positive ifψ〔A＋）⊂B＋． A⑧Bis also a syl国1etric algebra with an involuti㎝ such that 〔a） （a⑧b〕★＝at⑧b★ for a∈A．and b∈． B． （b）（A㊦B）＋・｛f・L〔A⑧B，．￠）；f（xxつ≧0』〔。。AXB）｝． 1． Cc呵）1ete positivity in the categoτy of symetric algebras． ［1．1］ DEFINITION． SuppOse that A and B aTe symmetric algd）ras． LetMn㈹d・n・t・the a19・bra・f・11 n刈mt・ice・・ver￠・Th・皿it。f叫（￠）
is denoted by ln． Alinear mappingτ：A→Bis said to be c（垣唖）1etely Positive if fbr eadh non−negative integer n， ・n；τ⑧1n・Mnω→Mn〔B）i・p・・i輌・・whe・・Mn〔A）＝A⑧当〔の鋤d監〔B）＝B㊦Mn（の・
［1・2］㎜・ L・tAbea・）・・鵬t・i・・a19・b・a鋤x∈Mn（A）・
Then
（1・2・a）x・c“ is a sum°f【・i・j “］−t）？・鵬t・エces・砲ere ai・aj・A・ （1・2・b）［ai牢］・M。（A） i・PO・itiv・・砲・・e ai・aj∈A・ 〔i， j ＝ 1，2，一一一一，n〕台［1．3］ PROPOSITION． Then we have； （1．3．a） 〔1．3．b〕 （1．3．c） （1．3．d） 〔1．3．e） Strppose that A and B are synmetric algebras．
ψis c卿1・t・1y p。・itiv。 ifψ・ゴ． ’
τ is completely positive if T∈ Hdm．〔A， B）． 、 IfπξHom〔A， B）， vξ B andエfτis defined byτ〔a｝＝v★rr（a）v， thenτis carrrpletely positive f Assume thatτ：A−≒＞B is co頓pletely pOsitive and that φ：A−一＞Bis defined by．φω＝τ（・vxvk）fbr a fixea v．q・A． Then φ is co田Pletely POsitive． Assume that （1） Kis a Hilbert space and H is a closed sdbSpace of K． （2）π．∈Hbm〔A，氾〔均）withπ〔1）＝1．・ （3）v∈磐（H，．K） Th㎝ifτis defined byτ（x）＝ vkr（x）v，τis co叩1etely positive． ［1． 4］ （1．4．a） PROPOSITION． （1．4．b〕 Proof of 〔1．4．a） obviously c（xrrl［）1etely pQsitive and thatτ： negatiVe integer， （φ・〕n＝（φ⑭1n）〔・⑧1n）＝φn・パSinceφ
andτ are positive， we have the conclusion． n t「°°f°f（1・4・b）：Fi・・t…b・’・rv・th・t if［・ij］∈叫（E）…d［βij］・M。（F）・・then
・ ilj ”・j⑧βij∈〔E玄⑧F）・ ’
Where E and F are s）㎜etric algebτas． Now use the c㎝plete positivity ofψand θ to dbtai工l the result． SymmetTic algd）ras fb㎝．a category with con輻）1etely POsitive 皿aps as morphisms．Assume that
〔1） A，B，C and D aTe syロ皿etric algel）ras． 〔2） ψ：A→B and θ：C→D are c（卿1etely positive．Th㎝
ψ⑧θ：A⑧C→B．⑧D is positive． ： Let A be a symmetric algebra． The identity of・Ais ． SUppose that A，B and C a re synmetric algd）ras A→Bandφ：B’→C are c（珊4）1etely positive． Fbr eadl non− 〉 、．．44
K．KUWANO
［1． 5］ PROPOS工TION． ．Assme that 〔1〕 Ais a sy皿皿etric algebra with unit・ 〔2） His a Hilbert space over￠． 〔3） τ：A→田（H）is co㎎）1etely positive withτ〔1〕＝1． Then there exists a pair〔K，π〕of a Hi1］bert space l（containing H as a closed sub・pace ・nd・h㎝・m。rPhi・m・…u・；h・th・t咽is a・）rmmet・ic ・19・bra・f c1・・able operators in K with the co㎜）n dense domai皿． Mbreover， we have τ〔a）＝Pπ（a） onH 〔a∈A）， 硫ere P is the proj ection of K onto H・ ［1．6］㎜（． If A i・a・C・・−a19・bτa， th・n咽i・aut・m・tically in 8〔H） （Steinspring ［1］）・ 2． Total pos itivity in the category of sy㎜etric monoids． S（M） honx）tu）rphism satisfying the following conditions； fbr any symmetric algel）ra A with a s）㎜etric homomo町phismψ：M→A， there exists a symmetric algebra−homomorphismφ：S（M）→A such that the above diagram．co㎜tes and S（M） is generated by ρCM〕． ［2．1］皿FINITI〔〕N． L・t M・・nd・L b・・）㎜・t・i・m・・ids・A・m・pPing・f：M→L is called totally positive if S〔f〕 ：S〔M）→S〔L） is co1耳pletely positive． リ ロ12．2］PROPOSITION． Fmctions of the following types are totally posltlve．
〔2．2．a〕 a sy皿netric dlaracter 〔2．2．b） asymmetric homolno蛸phism between M and L ［2．3］DEFINエTI（）N． L・t・M・・nd・L・b・ ・ym・t・i・m・n。id・・Th・n th・di・ect P・・du・t M×L・f M・・nd・L f・rm・a・ynm・t・i・m・n・id with㎜1tiplicati・n a・d lt−operation def血ed as fo11㎝s； （a〕 （X，y） 〔xl，y．） ＝ （XX’，yyt〕 〔b〕〔X，y〕★＝〔xft， yft〕 MOI． denotes the symmetric monoid constructed fro皿M＞くL． 〆Let M， Mt t L and L’be s）㎜etricロPnoids． Letψ：、 M→1りθ
We obtain a new．map ψOθ：M◎． M．→LOL’defined by
〔ψOθ〕〔X，X’〕＝〔ψ〔X），θ（Xり）． ：M・→L，． ［2． 4］ PROPOSITION． 〔2．4．a） S（MOL〕 ＝S（M）．⑭S（L） 〔2．4◆b） S（ψOθ） ＝S〔ψ〕《8S〔θ） ．（2．4．c）Ifψandθare sy㎜etriゆ㎜mrp垣sms， then so isψOθ． 〔2．4．d） Ifψandθare totally positive， thenψOθis positive． P「°°f・f〔2・4・・〕・S・pPO・e th・t旬・M→S・Pt）・・d ％・L→Sa）ar・−can・血・al maps・ We define a sy皿etric hcnTK）morphismρ ：MOL→S OyD ⑧S〔L） byP（x・y）＝㌔〔・〕・⑧・PL〔y）・ ・
Given a s）mnetric algebra A with syロ皿etric homomorphismψ：MOI、→A， the linear mapψ：S〔M）⑧S〔L〕、一一一Acan be《1efined by φ（ρ（x， y）〕＝ψ〔x， y〕， that is 〔S（M）⑧S〔1、），ρ）is the corresponding algebra for M O L． ゴ Proof of 〔2．4．1）〕： Let ρbe as in the proof of （2．4．a）． Define ρ・by ρ’〔x’syり＝P’M・〔xりOO P’L・の・ 工hen we have （S（ψ） ⑧S（θ）〕ρ ＝ ρ．〔ψ 〇二θ）， 砲㎝ce ・ 〔S〔ψ〕 ⑧ S（θ））ρ ＝ ρ，〔ψ Qθ） ＝ S（ψOθ）ρ．ThUS
S〔ψ◎ θ〕 ＝ S（ψ〕 ⑧S〔θ）． ． Proof of （2．4．c）： obvious． ’ Proof of （2．4．d）： By 〔2．4．b〕，we have S（ψO θ） ＝S〔ψ）⑭S〔θ）． Now since S（ψ〕⑧ S（θ〕 is positive by 〔1．4．b）， the assertion fbllows． 、 3・ （bmplete positivity in the category of馴匝netric monoi白． 1．et A and B be sy㎜etric algd）τas． Letψ：A−＞B be c〈珊｛pletely positive and 1・tΦ：Sω→B b・th・h・m…XPhi・m indu・ed・by・tl・u・h th・tψ・ΦpA・ Nbte that ψis c6卿pletely positive if and only if｛p is c（卿1etely positive． ［3．1】 DEFINITION． Slrppose that M is a syn皿etric monoid and that A is a ’ s）抽metric algel）ra． Amappingψ：M→A is said to be completely positive if induced honK）morphism｛P：S（M）→Ais cq㎎pletely positive．46
K．KUWANO
The fb11（痂g Proposition can・．．1）e verified by direct calcuユati㎝・ ［3． 2］ PROPOSITION． ㎞ctions of the fbllowing’types aエre completely positive． （3．2．a） a symmetric dharacter ． 〔3．2．b） a positive s）rmmetric function − 〔3・2．c） a sy㎜∋tric ho瓜）morphisln flr㎝a symmetriC monoid to a syl皿etric algebra． In particular， the canonical s）㎜metric hormmorphism ρ：M→SCM）is corrplete工y positive． 13．3］ PROPOSITION． Let M be a symetric lぴonoid and let A be a syrmetric algd）ra． Ifψ：M−一∋レA is totally pOsitive， thenψis c（匝pletely positive． 西゜°f：N°te that if［αi」］∈M。〔Sω）is P・sitive・thenI［・誇］・・P・・i廊・靱ω
砲ere
・・ゴ1・善・6珠……一…i・」…一・・）
by・Pplying the・泌・・fa・t t。 th・㎜t・iX［S（ψ）・ij】∈M。〔S（A）〕鋤鋤the c（㎎）1ete POsitivity of S〔ψ） ： S（M）一＞S〔A〕，we Obtai皿thatψ is c（m互）1etely positive． ［3．4］ THEOREM． Assume that （1） Mis a s）屹tric monoid and H is a Hilbert space． 〔2〕 Amapτ：M→磐〔H）is corrrpletely positive with τ〔e〕＝1． Then there exist a Hilbert space K containing H as a closed sdbspace and a sy㎜etric homomoτphismπ：M＿＿〉π0りsuch that τ〔x）＝Pπ（x〕㎝H〔x∈M）， 砲ere p is the orthogonal proj ection of K onto H andπ（M） is a symnetric mDnoid of closable operators in K with the common dense domail1・ We shal1 now introduce an adniss ible condition on M． ［3．5］ DEFINITION． Asy㎜etric mDnoid M is said to be poSitively boundedif
Strp｛ψ（x★x）；ψ∈y〔M〕｝〈。。 for each x∈M．［3． 61 THEOREM● In addition to the assuロrpticEls in 【3． 4］， sqlrp《）se that｝ Mis positively bo岨ded．「『hen there exists a MlbeTt Space K ccmtaiエ辻ng H as a closed subspace with a sy皿retric hr nonK）rphisinπ ：M→〉 ＄（K）such that τCx）＝Pw（x） onH fOr eaCh x∈μ，砲eTe P denotes the pTojecti㎝of K㎝to H・ Proof： If （K，π） is the dilatiαn given in Theoren 【3． 4］， th㎝the positive bOundedness of M il口plies the contiエ皿ity ofτand the b◎undedness ofπ（ix）fbr eaCh xξM． Details and so孤e apPlications・will aPpeaヱin fbrth−c《ぬiロ9 Papers・ The author wishes to e）qpress his gratitude to Prof． T． S垣bata fbT his
