On Minimal  -Completions of C*-Algebras

ON MINIMAL ^-COMPLETIONS OF C*-ALGEBRAS J. D. MAITLAND WRIGHT Let si be an arbitrary C*-algebra (with identity) and let A be the space of selfadjoint elements of si. We recall that A is a partially ordered real vector space with positive cone {a*a: aesi}. If each upper bounded monotone increasing sequence in A has a least upper bound in A, then si is said to be monotone a-complete. It was shown in Wright [12] that each monotone <r-complete C*-algebra is the quotient of its Baire* envelope by a two-sided cr-ideal. It is proved here, in Theorem 3.5, that any C*-algebra si (with identity) can be embedded in a minimal monotone <7-complete C*-algebra and, further, the minimal c-completion of si is the quotient of the Baire* envelope of si by a certain two-sided c-ideal. 1. Some preliminary results on C*'-algebras Throughout this paper si is a C*-algebra with identity / and A is the partially ordered real vector space of self-adjoint elements of si. Then A is an order-unit space with order-unit / and the order-unit norm on A coincides with the C*-norm. Let J be a closed subspace of A and l e t « / = J+iJ, then y is a proper closed two-sided ideal in si if, and only if, J is a proper near lattice ideal in A, see [1] and [2]. We recall that if / is a left (or right) ideal of si and / is self-adjoint (that is, / = / * ) then / is necessarily a two-sided ideal. LEMMA 1.1. Let fl be a proper closed self-adjoint ideal of si and letq be the quotient map of si onto si//. Let {hn} (n = 1, 2, ...) be an upper bounded monotone increasing sequence of self-adjoint elements of si// and let b be an upper bound for this sequence Then, there exists a B in A such that q(B) = b and there can be found a monotone increasing sequence {Hn} (n = 1, 2,...) in A such that B is an upper bound for this sequence andq(Hn) = hnfor n = 1, 2 , . . . . This follows by successive applications of Proposition 5 of [9]. si is said to be monotone a-complete if, whenever {an} (n = 1, 2, ...) is a monotone increasing upper bounded sequence in A then the sequence possesses a least 00 upper bound V an in A. When si is monotone o--complete a (vector) subspace E of I A is said to be a cr-subspace if, whenever {an} (« = 1, 2,...) is a monotone increasing 00 00 sequence in E with least upper bound V an in A, then V an is in E. I I The following lemma reduces, when / e E, to an elementary result on order-unit spaces (see, for example, Lemma 1.2 [11], or, for a C*-algebra approach, Lemma Received 28 January, 1973; revised 5 July, 1973. PULL. LONDON MATH. SOC, 6 (1974), 168-174] ON MINIMAL (T-COMPLETIONS OF C*-ALGEBRAS 169 2.4.2 [13]). As we wish to apply the result to ideals it is necessary to remove this restriction. For the convenience of the reader, we give a simple self-contained argument. LEMMA 1.2. Let sf be monotone a-complete and let E be a a-subspace of A such that, for any integer r ^ 1, whenever xeE then x2reE. Then E is a closed subspace of A. 00 When {cr} (r = 1, 2,...) is a sequence of positive elements of E and £ c r is conln \ 1 vergent then{Xc IT r > (« = 1,2, ...) is a monotone increasing sequence in E whose least upper bound in A is Z cr- Since £ is a <r-subspace of A it follows that Z cr is 1 1 in E. Let aeE such that ||a|| ^ 1 then, by the binomial series expansion and the Gelfand-Naimark theorem, 00 M= Z r= l For each r, {1 — (1 —a2)r} is a finite sum of even, positive, powers of a and so is in E. Hence, whenever aeE then \a\ eE. 00 Let {br} (r = 1,2, ...) be any sequence in E such that Z 11^11 < °°- Then I {br + \br\} (r = 1, 2, ...) is a sequence of positive elements in E and, as ||M-|6r|||=2||6P||, 00 00 00 I I I Z (br+ \br\) is in E. Similarly, £ |£rl is m £• So Z br, being the difference of elements of E, is in E. Thus £ is a Banach space and so a closed subspace of A. When si is monotone a-complete, an ideal / of *R/ is said to be a a-ideal if, and only if, y is a self-adjoint (and hence two-sided) ideal of si such that # n /I is a ff-subspace of A. It follows from Lemma 1.2 that every tr-ideal </ is a closed twosided ideal. Let .s/, ^ be monotone a-complete C*-algebras with identity. Then a function h: si -> 08 is said to be a a-homomorphism if A is a C*-homomorphism mapping the identity of si to the identity of 28 and such that, whenever {an} {n — 1, 2, ...) is an upper-bounded monotone increasing sequence in si then Clearly the kernel of a tr-homomorphism is a a-ideal. The next proposition states that, conversely, every c-ideal is the kernel of some a-homomorphism. 1.3. Let %> be a monotone a-complete C*-algebra with identity. be a proper a-ideal of %> and let q be the quotient homomorphism of %> onto Then # / , / is monotone a-complete and q is a a-homomorphism. PROPOSITION Let f 170 J. D. MAITLAND WRIGHT See Lemma 2.13 of [14]. We now introduce some more notation and recall some standard results in C*-algebra theory. Let X be the state space of si and let dX be the set of extreme points of X. Let H be the universal representation space of si and identify si with its image under the universal representation into £?(H). Let si" be the von Neumann envelope of si in ££(H). Then, see §12 Dixmier [4], si" may be identified, as a Banach space, with the second dual of si. Further, the self-adjoint part of si may be identified with A(X), the space of real affine continuous functions on X, whereas the self-adjoint part of si" may be identified with A"{X), the space of all bounded affine functions on X. The weak operator-topology of «£?(#) induces the same topology on si" as the weak (si", si')-topo\ogy ( s e e §12 Dixmier [4]). We shall use these identifications in the sequel without further comment. 1.4. Let si, <& be C*-algebras with identity and let k : si - » # be a C*homomorphism. Let k" : si" -> (€" be the second adjoint ofk. Then k" is a continuous map of si", equipped with the weak (si", si')-topology, to %>", equipped with the weak (%", #')•topology\ Also k" is a C*-homomorphism. LEMMA This is a straightforward application of §12 Dixmier [4]. Before stating the next lemma we must define the Baire* envelope of a C*algebra. Let /lo°° be the intersection of all cr-subspaces of the self-adjoint part of si" which contain A. Then each upper bounded monotone increasing sequence in y40°° converges in the weak operator-topology to a point of AQ™. Let J/Q 0 0 = /4000 + i i 0 c o then it is proved by Pedersen [7] that J ^ 0 ° ° is a C*-subalgebra of si" and he defines J^O00 to be the Baire* envelope of si. LEMMA 1.5. Let si, # be C*-algebras with identity and let si^, #0°° be their respective Baire* envelopes. Let kox : si^ -* #0°° be the restriction of k" to sio™. Then k0™ is a a-homomorphism of JZ/O°° int0 ^o 00 - See Proposition 4.2 [15]. 2. Monotone a-completions of C*-algebras For simplicity, we consider only C*-algebras with an identity element. Definition. A monotone a-completion of a C*-algebra si is a pair (#, k) where 92 is a monotone a-complete C*-algebra and k : si -> # is a C*-monomorphism with the following properties. (1) Whenever {an} ( « = 1,2,...) is a monotone decreasing sequence of selfOQ adjoint elements of si with greatest lower bound 0 then /\k(an) = 0. (2) # is a-generated by k[si], that is, # is the smallest *-subalgebra of # which contains k[si] and is such that its self-adjoint part is a ff-subspace of # . ON MINIMAL ^-COMPLETIONS OF C*-ALGEBRAS 171 Definition. A minimal o-completion of the C*-algebra sJ is a monotone ccompletion {$}, h) such that, whenever (#, k) is a monotone c-completion of s# then there exists a c-homomorphism / of ^ into # such that A; = //z. t •/ I An elementary argument shows that if (&u1^) and (^2> ^2) a r e minimal ^-completions of sf then ®x and ^ 2 a r e isomorphic, that is, when a minimal ^-completion exists it is unique " up to isomorphism ". Of course, it is not obvious that a minimal a-completion does exist. The remainder of this paper is devoted to proving that each C*-algebra possesses a minimal a-completion. LEMMA 2.1 (Choquet). Let K be a compact convex subset of a Hausdorff locally convex topological vector space and let dK be the set of extreme points of K. When dK is equipped with the relative topology induced by K then dK is a Baire space. A proof of this result may be found on page 49 of Alfsen [1] or on page 355 of Dixmier [4]. Let Jt+{s0) denote the cone of positive self-adjoint elements m in A"{X) such that {xedX : m(x) > 0} is a meagre subset of dX. Further, let Jf(s4) be the complex linear span of Jt + (s#) and let J(s#) be the intersection of Jf(sf) with S/Q™, that is J(sf) = Jf{sf) n J / 0 ° ° . LEMMA 2.2. J(sf) is a a-ideal of sto™ and stf n J{s$) = {0}. By Proposition 5 of Wright [12], S(s#) is a (two-sided) self adjoint ideal of S/Q*1 and si n J(s#) = {0}. By Lemma 1 of [12], JV{S#) is sequentially closed in the weak operator-topology of jSf(if). So any upper bounded monotone increasing sequence in J(sJ) = Jf{s4) n ^ 0 °° converges in the weak operator-topology of to an element of J(st). Thus J^(J^) is a tr-ideal of J/Q00LEMMA 2.3. When <& is a monotone o-complete C*-algebra then there exists a a-homomorphism p from ^Q° onto %> whose kernel is <?(<#) and is such that p{c) = c for each cetf. This is Corollary 7 of Wright [12]. Let SC0(si) be the set of all m in Jf+(sf) such that there exists a monotone decreasing sequence {bn} (n = 1, 2,...) in A where m(x) = infbn(x) for each xeX. Let .S?(«flO be the smallest tr-ideal of ^ 0 °° which contains «Sfo(j^). Clearly 172 J. D. MAITLAND WRIGHT LEMMA 2.4. Let h be the C*-homomorphism of sJ into «S/ 0 °7J5?(.S/) defined by h(a) = a+£?(s/) for each aestf. Then {si $* I $£ (st), h) is a monotone o-completion of st. By Lemma 2.2, s/ n 3f(s/) c sJ n J(sf) = {0}. So h is a C*-monomorphism. Let q : $4^ -> S/Q^/SP(sf) be the quotient homomorphism. Then, by Proposition 1.3, q is a <r-homomorphism and j2/0°°/jSf (.s/) is monotone a-complete. Further, g (a) = A (a) for all crej^. Let {bn} (n = 1, 2,...) be a monotone decreasing sequence of self-adjoint elements of J ^ with greatest lower bound 0. Let b(x) = inf bn(x) for each xeX. By Theorem 2.3 of Wright [11], the set {xedX: b(x) > 0} is meagre and so be&(s/). Thus To complete the proof we must show that h[s/] c-generates s/ Let W be the smallest <r-subspace of the self-adjoint part of s/a"0/&(&/) such that h[A] c W. Let V = {ae/4 0 °°: g(a)e W) then K is a <r-subspace of Ao™ and / 4 c K Thus F = An™. It follows that q maps J/Q 0 0 into the complex span of W. But q maps ^ 0 ° ° onto st^lSBisd). So / » M does a-generate s4 2.5. Let si be a C*-algebra with identity then ( ^ Q 0 0 / ^ ^ ) , h) is a minimal a-completion of sf. THEOREM Let (#, k) be any monotone c-completion of si. Since # is monotone c-complete, it follows from Lemma 2.3 that there exists a cr-homomorphism p : #0°° -> ^ whose kernel is y ( ^ ) and such that p(c) = c for each c e #. By Lemma 1.5 the C*-homomorphism k: #/-*<& can be extended to give a (T-homomorphism /co°° : s#0°° -* ^o*- L 6 1 ^ ^ e an Y element of S?0(sf), then there exists a monotone decreasing sequence {bn} (« = 1,2, ...) in s/ such that b(x) = inf&B(*) for each xeX. So A:ow(6) = inf AroTO(ftn) = in When aeja/ such that a ^ bn for all n then {xe3X : a(x) > 0} is an open subset of the meagre set {xedX : b(x) > 0}. But, by Lemma 2.1, open meagre subsets of dX are empty and so a < 0. Thus 0 is the greatest lower bound in s/ of the sequence {bn} ( « = 1,2,...). Since (#, k) is a monotone a-completion of s4 it follows that Hence, by Theorem 2.3 of Wright [11], the set {xedX: infk{bn) > 0} is meagre. n So infk(bn) is in J5?o(^)- Thus A:o°° maps ^ f 0 ^ ) m *o So J^o(^) is contained in the kernel of the o--homomorphism pk^. Since this kernel is a a-ideal it follows that J£?(.s/) is in the kernel of p^ 0 °°. It follows by elementary algebra that there exists a C*-homomorphism/mapping into # such that when q is the quotient a-homomorphism of s/om onto ON MINIMAL (T-COMPLETIONS OF C*-ALGEBRAS then fq(a) = pkon(a) for all a e ^ o 0 0 - Hence, whenever 173 aesf, fh(a) =fq(a) = pko»(a) = pk{a) = A:(a). So /A = /:, as required. Let {bn} (« = 1,2,...) be a monotone increasing upper bounded sequence in sfowl&(s0) then, by Lemma 1.1, there exists an upper bounded monotone increasing Then sequence in JS/Q00, {£„} ( n = l , 2 , . . . ) , such that qBn — bn for each n. = pk0»(Bn). So = /<z(supJ3n). Since g is a a-homomorphism, Thus Hence / is a <r-homomorphism and the theorem is proved. COROLLARY 2.6. When st is monotone a-complete then S£(sf) — When s/ is monotone c-complete it is its own minimal a-completion. So References 1. E. M. Alfsen, Compact convex sets and boundary integrals (Springer, 1971). 2. and T. Bai Andersen, " Split faces of compact convex sets ", Proc. London Math. Soc, 21 (1970), 415-442. 3. E. B. Davies, " The structure of S*-algebras ", Quart. J. Math. (Oxford), 20 (1969), 351-366. 4. J. Dixmier, Les C*-algibres et leurs representations (Gauthier-Villars, 1969, second edition). 5. E. G. Effros, " Order ideals in a C*-algebra and its dual ", Duke Math. J., 30 (1963), 391-412. 6. P. R. Halmos, Boolean algebras (Van Nostrand, 1963). 7. G. K. Pedersen, " On weak and monotone o-closures of C*-algebras ", Commun. Math. Phys., 11 (1969), 221-226. , " Measure theory for C*-algebras ", Math. Scand., 19 (1966), 131-145. 8. 9. , "A decomposition theorem for C*-algebras ", Math. Scand., 22 (1968), 266-268. 10. R. T. Prosser, " On the ideal structure of operator algebras ", Memoirs Amer. Math. Soc, 45 (1963). 11. J. D. M. Wright, " Measures with values in a partially ordered vector space ", Proc. London Math. Soc, 25 (1972), 675-688. 12. , " Every monotone a-complete C*-algebra is the quotient of its Baire* envelope by a two-sided w-ideal ", / . London Math. Soc, 6 (1973), 210-214. 13. F. Combes, " Quelques proprtetes des C*-algebres ", Bull. Sc. Math., 94 (1970), 165-192. 174 ON MINIMAL (T-COMPLETIONS OF C*-ALGEBRAS 14. E. Christensen, " Non commutative integration for monotone sequentially closed C*-algebras " Math. Scand., 31 (1972), 171-190. 15. G. K. Pedersen, " Measure theory for C*-algebras III", Math. Scand., 25 (1969), 71-93. Department of Mathematics, The University, Reading.