ON THE CONJUGATE SPACE OF OPERATOR ALGEBRA MASAMICHI TAKESAKI. (Received June 1, 1958)

Transcription

1 ON THE CONJUGATE SPACE OF OPERATOR ALGEBRA MASAMICHI TAKESAKI (Received June 1, 1958) The purpose of this paper is to study the structure of operator algebras as Banach spaces tzat has been developed by J. Dixmier, Z. Takeda, S. Sakai and others. 1 is devoted to study the structure of the conjugate spaces of an operator algebra, in which a certain decomposition theorem on the conjugate space is proved with some applications. In 2, we prove the decomposition theorem of a homomorphism from an operator algebra into the other, corresponding to the decomposition of the conjugate space in 1. Then, using this result we study the continuity property of a homomorphism and find the alternative proof of those results which are discussed in J. Feldman and J. Fell [7]. The author wishes to express his hearty thanks to Prof. M. Fukamiya and Mr. S. Sakai for their many valuable suggestions in the presentation of this paper. 1. The conjugate space of operator algebra. We denote always by M* for the conjugate space of a Banach. space M and if M is a W*-algebra we write the space of all a-weakly continuous linear functionals on M by M*. Let M be a C*-algebra, then the second conjugate space M** of M is a W*-algebra and its a--weak topology coincides with a- (M**, M *)-topology by Z. Takeda [ii]. M** has further properties suchn as any *-representation of M on some Hilbert space has a unique a-weakly continuous extension to a *-representation of M**. Thus we call this W-algebra universal enveloping algebra of M and denote by M. Next, we define the operators La, Ra on M*, the conjugate space of C*-algebra M, for a E M as follows; <x, Lar>=<ax, q)> and <x, Rag><xa, c>for all xem, pem*. Then the following properties are easily verified. L(a+b)=XLa+L7, R(Aa+b)=XRa+Rb L(ar)=LLa, R(ab)=RaRb, where a and b are arbitrary elements of M, X and arbitrary complex numbers. We denote the family of all La (resp. Ra) by L(M) (resp. R(M)). Next we call a subspace V of M* left invariant (resp. right invariant) if it is invariant under L(M) (resp. R(M)). Especially, we call a two-sided invariant subspace invariant simply. Then we have the duality of left ideal and left invariant subspace in the following THEOREM 1. Let M be a C*-algebra, then there exists a one-to-one corres-

2 ON THE CONJUGATE SPACE OF OPERATOR ALGEBRA 195 pondene between the o-weakly closed left ideal (resp. right ideal) m of M and the closed left invariant (resp. right invariant) subspace V of M* such that >=V and V=m where V and m are the polar of V and m in M and M* respectively. Especially, if M is a W*-algebra, then there exists a one-to-one correspondence between the a-weakly closed left ideal (resp, right ideal) and closed left invariant (resp. right in ariant) subspace of M*. PROOF. Let V be a left-invariant closed subspace of M* and in=v Since LaV c V for all a E M, we have<ax, V>=<x, LW>=<x, V> =0 for all xem. Hence we have axem for all aem, xem. Therefore in is a left ideal of i, for in is o-weakly closed and M a-weakly dense in M. The converse correspondence and the second part of our theorem are clear by the above argu ments. This concludes the proof. Using the similar argument we shall study the maximal left ideal of a C*-algebra. Let M be a C*-algebra and (p a positive linear functional on M. We call a subset of M mc={xem:<x*x, >=O} the left kernel of p by the terminology of R. Kadison [9]. Then we have the following THEOREM 2. Let M be a C*-algebra, cv a pure. state on 1V1, then. M/mQ, the factor space of M by the left kernel m0 becomes a Hilbert space as quotient space and its norm coincides with the one canonically induced by. PROOF. (1) Case where M is a W*-algebra and a-weakly continuous: By the continuity of p, there exists a minimal projection a of M such that =M(1-e) where e is the carrier projection of gyp. Hence M/mc is algebraically isomorphic to Me by the natural correspondence. If we denote by x the element of M/mc corresponding to xeme, we get Ixl=inf[Ilx+yll; yen1]=inf[ilxe+y(1-e)ii;yem] Ixel=Ilxil>Ilxl Therefore, the mapping. x x is an isometry from Me onto M/mc. Now, we consider the canonical representation 7r of M on the Hilbert space 14 induced by p in the sense of I. E. Segal. Then there exists a cyclic vector s of H, such that <x, qi>=(rc(x),) for xem. Since n,(e) is the projection to the one-dimensional subspace of 14 spanned by, we have Ixel=Iex*xeJI2=IIc(e)1r(x)*r(x)n(e)I12 =((xe))f2p=iinc(xe)ill Moreover, rc(m) is the full operator algebra on 14 for rr is a a-weakly continuous irreducible representation. Therefore, the mapping x-f rq(x)c is an isometry from Me onto 14. Hence M/mc is isometric to 114 by the canonical correspondence. (2) General case: Let M be the universal enveloping algebra of M,

3 196 M. TAKESAKI then q is a a-weakly continuous pure state on M. Let m be the left kernel of q in M, then the factor space MJm, coincides with the Hilbert space canonically constructed by q. in case (1). Put V=m4, then V is a left invariant subspace of M* by Theorem 1 and V*=M f mq. Hence V is a Hilbert space, so that V is o(m*, M) closed by its reflexivity, and V=m. Thus we have (Mm,)*=V and (MJm)**=Mf m, that is, M Jm is a Hilbert space and its norm coincides with the one induced by p. This concludes the proof. Applying this result for irreducible *-representation of C*-algebra, we can give the alternative proof of the Theorem due to R. V. Kadison [9]. But we omit the detail. DEFINITION 1. A positive linear functional q on a W*-algebra M is called singular if there exists no non-zero a-weakly continuous positive linear functional /r on M such as Moreover we call a linear functional i on M singular too, if p is a linear combination of singular positive linear functionals as defined above. We denote by M* the space of all singular elements of M*. THEOREM 3. Let M be a W*-algebra, then the left (or right) invariant closed subspace V of M* is uniquely decomposed into the l-direct sum of its cr-weakly continuous part V (M* and singular part V M*, i. e. Moreover V as follows: V=(V(1M*) 0+1(V(1 Mt). M* and V l M* are written by the central projection z of M V M*-RZ0V and V f M*=R(120)V. PROOF. As M* is an invariant subspace of M*, There exists a central projection z0 of M such that M*=M(1-z0) by Theorem 1. Hence we have easily M*=Rz0M*. Next we shall show M*=R(lzo)M*. In fact, if cp is a positive linear functional of R(zo)M* and there exists a a-weakly continuous positive linear functional I such as p, then we have <x*xz0, r<x*xz q>=0 for all x EM, which implies Rz0 fr=0. On the other hand, we have R(zo)*=0 by the continuity of and the above argument, so that we have /r=0. Hence p is singular. Conversely, if p is singular positive, then Rz0qi=0 because Rz0q<cv and Rz0c E M* Hence we have M * R(1zo)M*, for M and R(zo)M* are spanned by their positive parts respectively. Therefore, we have M*=RZOM*, M*=R(170) M* and M*=M* 3tiM*. Moreover, since V is invariant under R 0 and R(1,0), one may easily verify that V=(V M*) 1i(VfM*). This concludes the proof.

4 ON THE CONJUGATE SPACE OF OPERATOR ALGEBRA 197 This theorem includes the decomposition of a finitely additive measure into the completely additive part and the purely finitely additive part in the commutative case due to Yosida-Hewitt X16]. Using Theorem 3, we can show the following generalization of the well known Dixmier's result. COROLLARY 1. Let M be a W*-algebra, then a bounded linear functional o on M is a-weakly continuous if and only if o is a-weakly continuous on every maximal abelian W*-subalgebra of M. PROOF. It suffices to prove only the sufficiency. By Theorem 3, it is sufficient to show that there exists no non-zero singular linear functional satisfying the hypothesis. Suppose there exists a non-zero singular linear functional p satisfying the hypothesis. Moreover we may assume, without loss of generality, that p is self-adjoint (i.e. p is real valued on the selfadjoint part of M). There exists a projection a of M such as<e, p> 0 (or<0). Hence we may assume <1, 60>0, considering the restriction of 60 on eme. Now, if there exists a non-zero projection a of M such that <f, p>0 for all non-zero projection f<e, LeRe p is a non-zero o-weakly continuous linear functional by its positiveness. This contradicts to the singularity of q. Therefore, for any non-zero projection a of M, there exists a non-zero projection f<e such as<f, p>0. Hence, if {ea} is a maximal family of orthogonal projections such as <ec v>0, we have Lea=1. Considering the maximal abelian W*- subalgebra generated by {e, we get <1,p>0 by the continuity of y on this subalgebra. This contradicts to the hypothesis <1,60>0. This concludes the proof. Applying this result, we can show the following generalization of Umegaki's result in the case of semi-finite type [14]. COROLLARY 2. Let M be a W*-algebra, then a subset K of M* is relatively o-(m*, M)-compact if and only if, for each maximal abelian W*-subalgebra A of M, the restriction of K on A is relatively o-(a*, A)-compact in A*. PROOF. It suffices to prove only the sufficiency. Since M is the linear span of the self-adjoint part of M, K is simply bounded on M, so that K is bounded by Banach-Steinhaus' Theorem. Hence, if we imbed canonically K into M* and denote the o(m*, M)-closure of K by K, K is o(m*, M)-compact. We shall show Kc M*. In fact, for each maximal abelian W*-subalgebra A of M, the restriction of K on A is contained in the o(a*, A)-closure of the restriction of K on A. Hence the restriction of K on A is contained in A* by the hypothesis. Therefore, for each pek, q is c-weakly continuous on A. Thus we have KcM* by Corollary 1, that is, K is relatively o-(m*, M)-compact. This concludes the

5 198 M. TAKESAKI proof. We conclude this section to prove the characterization of a singular positive linear functional in the commutative case. THEOREM 4. Let A be an abelian W*-algebra, 11 the spectrum space of A. Then, in order that the position linear functional on A induced by a Radon measure on f is singular it is necessary and sufficient that its support is non-dense. PROOF. Sufficiency is clear from the definition of singularity so that it suffices to prove only the necessity. Let F={Na} be a maximal family of mutually disjoint non dense Borel sets such as a(n)*0, then F is at most enumerable, for 1(U)=a(Na)<(11)<+c. Since an enumerable sum of non-dense sets is also non-dense in a hyperstonean space by Dixmier's result [3], N0=JNa is non-dense. Let N be any non-dense Borel set of 1, then (NN0)=0 by the maximality of F. Put <f, p>=<xnvf, i>for fec(1)a and j= -j,, where XNo is the characteristic function of N0i then any non-dense Borel set is Jh-measure zero, so that P2 is a normal measure by Dixmier's result [3]. Therefore, if the positive linear functional on A induced by tc is singular, we have=0, so that 1 has a non-dense support. This concludes the proof. 2. Homomorphism of operator algebra. We start with the following DEFINITION 2. A bounded linear mapping A from a W*-algebra M to another W*-algebra N is called singular if t8(n) c M*, where t9 is the transpose of 6. We get the following theorem corresponding to the decomposition theorem for the conjugate spaces of operator algebras. THEOREM 5. Let M and N be two W*-algebras and 7a*-homomorphism from M into N, then there exists a central projection z in the a-weak closure of n(m) such that if we define the *-homomorphisms r and r from M into N as follows: 7(x)=r(x)z 7(x)7(x)(1-z) for all xem, then ik is o-weakly continuous and r is singular. PROOF. We may assume, without loss of generality, that r(m) is a- weakly dense in N. Let zo be the central projection of M as in Theorem 3. rr is uniquelly extended to a a-weakly continuous *-homomorphism n from M onto N by A. Grothendieck [8]. If we put z=r(z0), then z is clearly a central projection of N and has the properties in our theorem.

6 ON THE CONJUGATE SPACE OF OPERATOR ALGEBRA 199 In fact, let q be an arbitrary element of N*, then for all xe M we have <x, t(>=<n-(x), >=<ir(x)z, q> =<r(xzo), p>=<xzo, tn(q)> =<x, Riot ir(w)> where tn is the transpose of rr, Since R M*=M* by Theorem 3, we have tn lem*, so that r1 is o-weakly continuous. By similar computation we have tn 2(q))=R(1zo)tn(q) for all 7 E N*, so that tn2(n*)cm*, i.e. nu is singular. This concludes the proof. Combining the above result with the characterization of a singular functional in 1, we study the cr-weak continuity of the *-homomorphism from a W*-algebra into another cr-finite one. LEMMA 1. Let A be an abelian W*-algebra, B a or-finite W*-algebra and n a *-isomorphism from A into B, then there exists a non-zero projection e of A such that n is a-weakly continuous on Ae. Moreover, the a-weakly continuous part n1 of n is a *-isomorphism form A into B and the singular part ir of n is a non- faithful *-homomorphism from A into B. PROOF. r1(0) is a a-weakly closed ideal, so that there exists a projection z of A such that n1(o)=az. If z*0, then r2 is a singular *-isomorphism from Az into B. Now, there exists a faithful a-weakly continuous state (p on B by a-finiteness of B. Put 1=tn9(p), then r is a singular positive linear functional on A. Hence there exists a non-zero projection e<z such as<e, r>=0 by Theorem 4, so that we have <i2(e), o>=<e, t71((i9)>=<e, >=0, which implies n2(e)=0. This contradicts to the faithfulness of 712 on Az. Therefore we have z=0, i. e, n1 is faithful on. A. Finally, there exists a non-zero projection e of A such as r2(e)=0 by the above argument, so that Aec it 1(0). Then, n and n1 coincide on Ae and n2 is not faithful. Therefore, n is a-weakly continuous on Ae. This concludes the proof. LEMMA 2. Let M be a a-finite W*-algebra, then there exists a maximal abelian W*-subalgebra Ao of M as follows: if a*-homomorphism r from M to another one N is a-weakly continuous on A0, then n is a-weakly continuous on M. Moreover, if M is of finite type or type III, any maximal abelian W *-subalgebra of M has this property. PROOF. It is sufficient to show that n is a-weakly continuous on all abelian W*-sub Agebras which are generated by the family of orthogonal projections. (1) Type Ia case: Let Ao be a maximal abelian W*-sub. lgebra of M, {e a family of orthogonal projections and A be a maximal abelian W*-sub-

7 200 M. TAKESAKI algebra containing {en}, then u*aou=a for some unitary u EM. From the assumption, the mappings x-uxu*-7(uxu*)=7(u)n(x)n(u*) -r(u*)i(u)r(x(u*)r(u)=n(x) are continuous on A, hence 7 is continuous on M. (2) Type II, case: Let Ao be any maximal abelian W*-subalgebra of M, then for any projection of M there exists a projection of Ao equivalent to this projection. Moreover, one easily verifies that for any projection pie (eea0, PEM) there exists also a projection q of A, such as pq<e. Therefore, for an arbitrary family of orthogonal projections -(p4 in M, we can get the family of orthogonal projections {q} in A such as p. Hence there exists a partial isometry u in M as u induces the equivalency for all p,z and q,6. Furtaermore we can choose u as a unitary of M by the finiteness of M. That is, up,u*=q,z for all n. Now we consider the mappings as follows: x-uxu*-7(uxu*)r(u)n(x}z(u*) for xem. The continuity of r on A0 implies that of n on an abelian W*-subalgebra generated by {pn}. This implies the o-weak continuity of r, for {pn} is an arbitrary family of orthogonal projections in M. (3) Type I. or IL case: Let {e,4} be a family of orthogonal finite projections such as den=1 and A0 an arbitrary maximal abelian W*-subalgebra containing them. For a projection p E M, we shall show that Ao contains a finite projection q as By the assumption on {e} there exists a number n such that c(p)et0 0 where c(p) means the central envelope of p. Hence we get a central projection z such as zp (1-z)p (1-z)eno and zp0 or (1- z)e*0. If(1- z)en0*0, then q=(1-z)eno is the desired one. If (1-z)e, =0 we have zp*0, which implies the existence of a finite projection e of A0 as zpe. Applying the argument in case (1) and (2) for a finite W*- algebra eme and its maximal abelian W*-subalgebra eao, we get a projection q E Ao as zp-q. Now suppose a family F such that F={gEA0q} for a finite projection p of M. The above argument shows that F is a non-empty inductive set, so that there exists a maximal projection q of F and p becomes equivalent to q. Therefore, for a finite projection such as p e for some projection e of Ao we get a projection q E A0, such as q<e considering with a W*-algebra eme and its maximal abelian W*-subalgebra ea0. Next, if p is an arbitrary projection of M there exists a family of orthogonal finite projections -(p,a} such as p=hence, we have a family of orthogonal projection {qn} of A0 such that pngn for all n. Put q=qn, then q E Ao and Therefore the same computation for M and A0 used above shows that pie, e E Ao implies the existence of a projection q E Ao such

8 ON THE CONJUGATE SPACE OF OPERATOR ALGEBRA 201 as pq<e. After all, the above arguments show that Ao contains a projection e with e(1-e). Take an arbitrary family {p,z} of orthogonal projections of eme, then (1- e)ao contains a family {q,a} of orthogonal projections such as qpn for all n. Let vn be a partial isometry of M that gives an equivalency of f and qn, that is, v*vn=p, and vnv*=q, Put u=(vn+u) +(1-L(pn+qn)), then u is a unitary and induces all equivalencies between {pn} and {q,1}. Considering the decomposition of r by this unitary u as in case (1) and (2), we get the 4-weak continuity of 7r on the abelian W*- subalgebra of eme generated by family {p,}. Therefore r is a-weakly continuous on eme. If we decompose n into its continuous part and singular one, the singular part vanishes on eme, hence vanishes on M because of the equivalency of a and (1e). This completes the proof of case (3). (4) Type m case: Take an arbitrary maximal abelian W*-subalgebra A0 and a projection p of M. Denoting by c(p) the central envelope of p we consider the maximal family {qn} of orthogonal projections of Ao whose central envelopes are orthogonal each other and covered by c(p). Put q=lq,,, then, by the maximality of {qn}, we have c(p)=c(q), which implies pq. Therefore the same computation as in case (3) shows the continuity of 7 on M if 7 is continuous on A9. This concludes the proof. REMARK. Though we restricted our arguments to the case of a-finite W*-algebra, the above proposition holds without the assumption of a-finiteness. We omit the proof because this is immaterial for our further arguments. THEOREM 6. Let 7 be a *-isomorphism from a a-finite W*-algebra M into a o-finite W*-algebra N, then the continuous part 1 of r is a *-isomorphism and the singular part 7r of ir is always non-faithful *-lzomomohism. PROOF. We can find a central projection z of M such as 7rj(0)=Mz. If z*0, xa is singular faithful on Mz. So we may assume, without loss of generality, that Ira is faithful on M. Let A0 be a maximal abelian W*- subalgebra in the sense of Lemma 2. By Lemma 1 there exists some nonzero projection a of A0 such that n-1 is a-weakly continuous on eao. Hence Ira is a-weakly continuous on eme as ea0 has the same property in eme as Ao does in M. Since n2 is singular we have 7a(eMe)=0. This yields a contradiction. Therefore 'irl is faithful and irz is not. This concludes the proof. The above discussions have some applications. We shall show in the following that some of the results of J. Feldman and J. Fell [7] are proved by this method. At first, we start with the lemma which is somewhat well known (cf. J. Calkin [1]).

9 202 M. TAKESAKI LEMMA 3. if the kernel of a *-homomorphism r from a a-finite, properly infinite W*-algebra M into a W*-algebra N is not a-weakly closed, then there exists a family of orthogonal projections in N with the cardinal of at least the continum. PROOF. By our assumption there exists a monotone increasing sequence {e,s} in m=7-(0) with e=/ enm. Take a family of orthogonal projections {pn} in M and partial isometries {vn} such that pn1, vvn=p and vnvn=1 for all n. Define the family {qn} of orthogonal projections by qn=vnevn, then we have qnen1+(em+1-en,)=jen=ent. Next, let {rn} be a countable set of all rational numbers. For any real number s we can choose an infinite subsequence of {r} so that they coverge to s. Correspond s to an index set {fl} of the above sequence {ra,}. We have, if ss for real numbers s and s, {n} (1 {n} as at most a finite set where {ni} and {n} are corresponding index sets of s and s respectively. Therefore, if we set q=qnz for a real number s where {n} is an index set corresponding to s, we get qs in and gsgs Emfor s *s. Hence the family {9r(gs)} of orthogonal non-zero projections of N is of the power of continuum at least. THEOREM 7. Let M be a finite factor oy a-finite, properly infinite W*-algebra, then any *-homomorphism 'ir from M into a-finite W*-algebra N is a-weakly continuous. PROOF. Since a finite factor M is simple, there exists no non-trivial non-faithful homomorphism on M. Therefore i is a-weakly continuous by Theorem 6. Let M b properly infinite and suppose the kernel of the singular part rz of ir is a-weakly closed, then there exists a central projection z of M such that,-(0)=m(1-z). Hence ira is a *-isomorphism from Mz to N, which is impossible by Theorem 6. Therefore we have z=0, i.e. 72=0. On the other hand, if i- (0) is not a-weakly closed, then N is not a-finite by Lemma 3, which is a contradiction. We get, after all, i-20. This completes the proof. COROLLARY. Let M be a finitejactoi or a-finite, properly infinite W*-algebra and N a W *algebra on some separable Hilberc space, then any bounded honiomorphism (not necessarily *-preserving) 7 from M into N is a-weakly continuous. PROOF. Let V be the closure of tr(n*) in M*, then V is a closed invariant subspace of M*. Since N* is separable by our assumption for N, V js also separable. Therefore, V M* is also a closed invariant separable

10 ON THE CONJUGATE SPACE OF OPERATOR ALGEBRA 203 subspace of M*. Hence there exists a central projection z of M such as (V1M*)=Mz and (V(1M*)* M(1-z). By the separability of V(1M* M(1-z) is a a-finite W*-algebra. Therefore, by Theorem 7, the *-homomorphism from M to M(1-z) such as x-x(1-z) for x E M must be a- weakly continuous. Hence V M*=0 and by the equality V=(V M*) +l(v(lm*) we get Vc M*. That is, is a-weakly continuous. REFERENCES [1] J. W. CALKIN, Two sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. Math., 42(1941), [2] J. DIXMIER, Les algebres doperateurs dans 1espace hilbertien, Paris (1957). [3] Sur certains espaces consideres par M. H. Stone, Summa Brasil, Math., 2(1951), [4] Formes lineairs sur un anneaux d'operateurs, Bull. Soc. Math. France, 81 (1953), [5] J. DIEUDONNE, La dualite dans les espaces vectoriels topologiques, Ann. Ec. Norm. Sup., 59 (1942), [6] J. FELDMAN, Isomorphisms of finite type II rings of operators,, Ann. Math., 63 (1956), [7] J. FELDMAN and J. FELL, Separable representations Math., 65 (1957), of rings of operators, Ann. [8] A. GROTHENDIECK, Un resultat sur le dual d'une C*-algebre, Journ. Math., 36 (1957), [9] R. V. KADISON, Irreducible operator algebras, Proc. Nat. Acad. Sci. U.S.A., 43(1957), [10] S. SAKAI, A characteraization of W*-algebra, Pacific Journ. Math., 6(1956), [11], On topological properties of tiv*-algebra, Proc. Japan Acad., 33 (1957), [12] Z. TAKEDA, Conjugate spaces of operator algebras, Proc. Japan Acad., 30 (1954) [13] On the representation of operator algebras, Proc. Japan Acad., 30 (1954), [14] On the represenatation 6(1954), of operator algebras II, Tohoku Math. Journ., [15] H. UMEGAKI, Weak compactness in an operator space, Kodai Math. Sem. Rep., 8(1956), [16] K. YOsWA and E. HEWITT, Finite additive measures, Trans. Amer. Math. Soc., 72 (1952), MATHEMATICAL INSTITUTE, TOHOKU UNIVERSITY.