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(1)jfr~**I*~~:ji]f~¥Ili1r NQ41, 2007:¥, pp.23-27 Research Reports of the School of Engineering, Kinki University No41, 2007, pp.23-27. A strong duality for separable C*-algebras IT Kiyoshi IKESHOJI. Abstact. It is shown that every separable C*-algebra can be reconstructed from the pure state space. Keywords: C*-algebra, positive linear functional, GNS construction, pure state, duality. In this paper, we shall continue our investigations of a duality for separable C*-algebras. Contrary to the previous dualities, we shall take the pure state space as the dual object of separable C*-algebras instead of the space of irreducible representations. As consequences, we shall show a strong duality for separable C*algebras.. 1. Introduction. The fundamental Gelfand representation theorem asserts that every commutative C*-algebra is isomorphic to the C*-algebra of all continuous functions on its spectrum vanishing at infinity. The dualities for non-commutative C*-algebras are considared as generalizations of Gelfand theory to the non-commutative case. There are two kinds of dualities for non-commutative C*algebras. We call a duality strong duality that the dual object consists of only irreducible representations. On the other hand, we call a duality weak duality that the dual object contains not only irreducible representations but also reducible ones. It was Takesaki's weak duality for separable C*-algebras [6] that initiated the dualities for non-commutative C*-algebras. Bichteler generalized Takesaki's weak duality to the nonseparable case [2]. In [1], Akemann-Shultz showed a first semi-strong duality for every C*-algebras. Fujimoto interpolated Akemann-Shultz's results to establish a strong duality for every C*algebras using his CP-convexity theory [4]. In [5], we showed a strong duality for separable C*-algebras within the framwork of the representation theory.. 2. GNS construction. In this section, we shall review the GNS (Gelfand-Naimark-Segal) construction which is an essential role of our investigations. Our general reference on this subject is [3]. A linear functional W over a *-algebra A is defined to be positive if. w(a*a). 2:. 0 for all a E A.. If WI, W2 are positive linear functionals we write 2: W2 whenever WI - W2 is positive and we say that WI majorizes W2. A positive linear functional W over a C*algebra withllwll = 1 is called a state. A state w over a C*-algebra is defined to be pure if the only positive linear functionals majorized by ware of the form AW with 0 :S A :S 1.. WI. Department of Mechanical Engineering, School of Engineering, Kinki University. 23.

(2) 24. First of all, we shall review the well-known usefull properties of positive linear functionals over a *-algebras.. The operator 1rw (a) is obviously linear and since it holds that l1 1rw(a)(TJw(x))112 = (TJw(ax) ITJw(ax)). Lemma 2.1 Let w be a positive linear functional over a *-algebra A. It follows that. = w(b*a),. (1) w(a*b). = w(x*a*a x) :::; lIaWw(x*x). and. = II a 11211 TJw ( x ) 112 ,. (2) Iw(a*b)12 :::; w(a*a)w(b*b) for all pairs 1rw ( a). a, bE A. Hereafter A denotes a separable C*-algebra with a unit and A+ the set of all positive linear functionals over A. Lemma 2.2 For each w E A+, it follows that. (1) w(a*). (2). Iw(a)1. = w(a), 2. :::;. lI1rw(a)1I :::; lIall for all a E A. So 1rw (a) extends to a bounded linear operator, also denoted by 1rw(a ), on 1lw. The map, A :3 a I---t 1rw E fJ3(1lw) , is indeed a representation of A, where fJ3(1lw) denotes the full bounded operator algebra on 1lw. There exists a unique vector fw E 1lw such that. w(a*a)lIwll,. w(a). (3) Iw(a*ba)1 :::; w(a*a)lIbll for all a, bE A,. (4) IIwll=sup{w(a*a): (5). lIall=l}. =. Let {1r, 1l} be a non-degenerate representation of A and a vector in 1l. Then a positive linear functional w over A can be deduced as follows;. e. forallaEA.. = l.. Let consider this converse. For each w E A+, let define N w by. N w = {x : x E A, w(x*x). = O},. then N w becomes a left ideal of A. Let denote the coset x + N w by TJw (x) in the quotient space AIN w for each x E A. Let define. (TJw(x) I TJw(Y)) := w(y*x). for all x, yEA,. then it provides an inner product on AINw • Let 1lw denote the Hilbert space obtained by the completion of AIN w with respect to this inner product. For any a E A, let 1rw (a) denote the left multiplication operator by a on AINw , i.e.,. 1rw(a)(TJw(x)) := TJw(ax). for all a EA.. 1lw.. A representation of A is a *-homomorphism of A into the full bounded operator algebra on a Hilbert space. A representation {1r, 1l} of A is called non-degenerate whenever 1r(A)1l 1l holds.. w is a state if lIell. = (1rw(a)ew lew). The vector ew is obtained by ew := TJw (lA) in. and. Ilwll = w(l).. w(a):=(1r(a)fle). is bounded on A INwand it follows. for all x EA.. After all it follows that for each positive linear functional w over A, there exists a cyclic representation {1rw, 1lw, ew} of A such that. The representation {1rw, 1lw, ew} is determined up to unitary equivalence. The construction of {1rw, 1lw , ew} from w E A+ is called the GNS construction. At the end of this section, we shall review the next well-known theorem which plays an important role in our investigations. Theorem 2.3 Let w be a state over A and {1rw, 1lw, ew} the associated cyclic representation of A. Then the following conditions are equivalent: (1) {1rw,1lw} is irreducible; (2) w is pure; (3) w is an extremal point of the set S(A) of the state over A.. 3. Dual object Let H denote a fixed separable Hilbert space and Rep(A : H) the set of all non-zero representations of A on H. For each 1r E Rep( A : H), let H 7r denote the essential subspace of 1r, i.e.,the closure of 1r(A)H and let P7r denote the projection ofH onto H 7r'. If {u, K} is a representation of A on the closed subspace K of H, let identify.

(3) 25. A strong duality for separable CO -algebras IT. {O",I<} with {O" EB OKJ.., J{ EB J{ 1..} , where OK J.. is the zero-representation of A on the orthogonal complemenet J{ 1.. of J{ in H. By the separability of A, for each w E S(A) there exists a partial isometry U w of H onto 'Hw such that the representation U. ,where Il, v are probability Borel measures on the standard measure spaces r, Z and w-y E P(A) Il - a.e., w( E P(A) v - a.e., then it follows that. w*{7rw, 1lw}uw belongs to Rep(A: H), and. pu"".1-l w U w =. Uw. *u w , denoted by p(w).. Hence we obtain a surjective map the state space S(A) as follows;. ~. defined on. <P: S(A) 3 w I-t U w*{7rw, 'Hw}u w E Rep(A: H). According to [2], let furnish Rep(A:H) with the topology of pointwise convergence with respect to the weak operator topology on ~ (H). S(A) is now endowed with the weakest topology such that <P is continuous. Unfortunately, at present we have no intrinsic expression of this topology. Let denote P(A) the set of all pure state of A. Since A is unital and separable, the state space S(A) is metrizable convex weakly*cpmpact subset of A * and the set of all extremal points of S(A) is precisely identical with P(A). We shall regarde the pure state space P(A) with the topology inherited by S(A) as the strong dual object of A. Let denote Irr(A: H) the set of all 7r in Rep(A : H) such that {7r, 1£11"} is irreducible. Then the restriction <P p of <P to P (A) is a surjective map as follows;. <Pp : P(A) 3 w I-t U w*{7rw, 'Hw}u w E Irr(A : H). 4. A strong duality In this final section, we shall show that A is isomorphic to the C*-algebra of continuous functions on P(A).. Definition 4.1 Let P(A) denote the set of all ~(H)-valued functions x defind on P(A) satisfying the following conditions;. (a) x is continuous with respect to the weak operator topology on ~ (H),. (b) p(w )x(w )p(w) = x(w) for all w E P(A), (c) sup{llx(w)11 : w E P(A)}. < 00. and. (d) If it holds that. ["i. <f1p. (w~ ) dl'CY) =. 1'". If A is of type- I, condition (d) can be removed. In P(A) let endow with a normed* -algebraic structure as follows; For all a, j3 E C, and x, y E. (1) (ax. + j3y)(w). (2) (xy)(w). :=. :=. ax(w). P(A). + j3y(w) ,. x(w) y(w),. (3) x* (w) := x(w)* for all w E P(A) and. (4) IIxll := sup{ {x(w)1I : w E P(A)}. Proposotion 4.2 P(A) becomes a C*-algebra with respect to the above operations. Proof. It is easily seen that P(A) becomes a normed* -algebra with IIx*xll = IIx1l 2 , so we have to prove only the completeness. Let {xd be a Cauchy net in P(A). Then for each w E P(A), the net {Xi(W)} in ~(H) converges uniformly to an operator y(w) and this convergence is uniform with respect to w E P(A) by (4). So it is rather obvious that ~(H)-valued function y defined on P (A) satisfies the conditions (a),(b),(c) of Definition 4.1. Suppose that the hypothesis of the condition (d) of Definition 4.1 holds, then since it holds that. £'" y(w~) dl'CY) -1'" Y(wd dl/«() II $ II £'" Y(Wry) dl'CY) - £'" X; (Wry) dl'CY)1I + II £'" (Wry) dl'CY) -1'" x;(wd dl/(Oll II. Xi. + Ill'" x;(wd dl/«(). :; £. -1'". y(w(} dl/(Oll. Ily(w-y) - Xi(W-y) II dll("Y). + l"xi(w d. - y(wdll dv((). ::; 211Xi - yll ~ 0,. <f1p (wd dl/( (). it follows that y satisfies the condition (d) of Definition 4.1. Therefore y belongs to. P(A) and.

(4) 26. this implies that is a C* -alge bra.. .P(A). ~(w). P(A). is complete. Hence. 0. Proposition 4.3 A can be imbedded in. = p(w)(ab) = p (w) (a)p (w)(b). = a(w)b(w). P(A).. = (ab)(w),. Proof. For each a E A, let define ~(H)-valued function a defined on P(A) as follows; A. a * (w). a(w) := p(w)(a) for all w E P(A). By the definition of the topology on P(A) and. Rep(A:H), it follows that. for all w E P(A). Hence it follows that the map;. {Wi} converges to w in P(A). ¢:>. {p(wd} converges to p(w) in Rep(A:H).. ¢:>. {p (wd (a)} converges weakly to p(w)(a) in ~(H) for each a E A.. ¢:>. {a(wd} converges weakly to a(w) in. ~(H).. This implies the continuity of a. Since it holds that. p(w)a(w)p(w). = w*uw1l"w(a)uwuw*uw = w*11"w(a ) w =a(w), U U. = p ( w )( a *) =p(w)(a)* = (a)*(w). U. a satisfies the condition (b) of Definition 4.1.. A :3 a. = lIall· a satisfies the condition (c) of Definition 4.1. If the hypothesis of the condition (d) of Definition 4.1 holds then it follows that for all a E A. a E .P(A). is an isometric *-isomorphism. This completes 0. ~eproo£. Now we are in the position to prove our strong duality theorem. Theorem 4.4 A separable unital C*-algebra A is isomorphic to the C*-algebra .P(A).. Proof. By Proposotion 4.3, the rest of the proof is to show that the map, a f---7 a, is surjective. For each x E P(A), let define x as follows;. -. x(p(w)) := x(w) for all wE P(A).. Since it follows by (4) Iiall = sup{lla(w)11 : w E P(A)} = sup{111I"w(a)11 : w E P(A)}. I--t. Since the map, w f---7 p (w), is a surjection from P(A) to Irr(A:H), it follows that x is a ~(H) valued function on Irr(A:H). Now we are going to verify that for each x in. .P(A), x satisfies the conditions (a), (b) ,( c) and (d) of Definition 1 in [5]. p(w) x(p(w)) p(w). = p(w) x(w) p(w) = x(w) = x(p(w)),. By the definition of a, we have. by (b) of Definition 4.1 and the definition of X. Irr(A:H):3 p(w) f---7 x(p(w)) is continuous as the proof of Proposotion 4.3. Ilxll. This implies that a satisfies the condition (d) of Definition 4.1. After all we get that a belongs to .P(A) for every a E A. Let a, f3 E C and a, b E A, then it holds that. -. (aa + f3b)(w). = p(w)(aa + f3b) = ap(w)( a) + f3p (w)(b) = aa(w) + f3b(w) = (aa + f3b)(w),. = sup{llx(p(w))11 : p(w) E Irr(A : H)} = sup{ {x(w)11 : w E P(A)} = Ilxll,. by (c) of Definition 4.1 and the definition of Ilxll. Let 11" E Rep(A : H) and. be irreducible decompositions of 11" over Irr(A:H) in the sense of the definition in [5]..

(5) A strong duality for separable C'-algebras II. Let define P(A)- valued fields on Zi as follows; w(. E ~p -1(1l"1()) for (E Zl,. 0"(. E ~p -1 (1l"2 ()) for (E Z2.. Then it follows that Ef). (. lZl. "Ef). ~p(wd dJ-Ld() = ( ~p(O"d dJ-L2().. lZ2. By (d) of Definition 4.1 and definition of have that. x we. Consequently x belongs to C[Irr(A:H)] of Definition 2 in [5] for each x E ..P(A). Using Theorem in [5], we can conclude that there exists unique element a of A such that for all w E P(A) x(~p(w)). = ~p(w)(a) = a(w).. --. Therefore the map; A :3 a ~ a E P(A), is surjective and this completes the proof. 0. references. [1] C.A.Akemann and F.W.Shultz, Perfect C*algebras, Memoirs Ams. Math.soc.,326,(1985). [2] K .Bichteler, A generalization to the nonseparable case of Takesaki's duality theorem for C*-algebras, Invent. Math.,9,(1969) ,89-98. [3] O.Bratteli and D.W.Robinson, Operator algebras and quantum statistical mechanics 1, Springer-Verlag, (1986) [4] I.Fujimoto, Gelfand-Naimark theorem for C*-algebras, Pacific J. Math.,184,(1998),95119. [5] K.lkeshoji, A strong duality foy separable C*-algrbras, RRSE. Kinki Univ.,37,(2003),6973. [6] M.Takesaki, A duality in the representation theory of C*-algebras, Ann. Math.,85,(1967), 370-382.. 27.

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