Recent Developments in the Theory of Unbounded Derivations in C*-Algebras

Transcription

1 Proceedings of the International Congress of Mathematicians Helsinki Recent Developments in the Theory of Unbounded Derivations in C*-Algebras Shôichirô Sakai* 1. Introduction. The study of derivations in C* -algebras is making great strides in recent years. It is divided into two major steps. The first is for bounded derivations ; the second for unbounded derivations with dense domain. At the early stage, mathematicians devoted their effort to the study of bounded derivations. This is understandable, because bounded derivations can be more easily handled than unbounded ones so that one can expect a beautiful theory as mathematics and also knowledge on bounded derivations may contribute to the study of unbounded derivations. In fact, the study of bounded derivations is now attaining maturity. On the.other hand, the study of unbounded derivations occurred much later and was initially motivated by the problem of the construction of dynamics in statistical mechanics. Soon it became apparent that the work of Silov [26] also has strong influence on the study of unbounded derivations in commutative C*-algebras. For bounded derivations, the main theme is when they are inner. On the other hand, for unbounded ones, it is rich in variety, because they are closely related to dynamical systems in quantum physics and differentiations in manifolds. In this, I would like to give a very brief survey of recent developments in the theory of unbounded derivations in C*-algebras. Let 91 be a C*-algebra. A linear mapping b in 91 is said to be a ""-derivation in 9Ï if it satisfies the following conditions: (1) The domain 9(b) of b is a dense *-subalgebra of 91; (2) b(ab) = ô(a)b+ab(b) (a 9 be@(b)); (3) S(a*) = S(a)* (ae9(8)). If 9(S) = % then b is closed so that by the closed graph theorem it is bounded [20]. Therefore the study of everywhere defined derivations on 91 is equivalent to the study of bounded derivations. * This research was partially supported by the National Science Foundation of the United States.

2 710 Shôichirô Sakai 2. Closability. If 9(b)Ç^% then b is not necessarily closable [5]. An element x of the self-adjoint portion 9(b) s of 9(b) is said to be well-behaved if there is astate cp x on 91 suchthat!?,(*)! = x and q> x (b(x))=0. Let W(b) be the set of all well-behaved elements; then it is dense in 9Ï S. b is said to be well-behaved (quasi well-behaved) if W(b)=9(b) s (the interior W(b) is dense in 9(b) s ). Any (quasi) well-behaved ""-derivation is closable and its closure is again (quasi) well-behaved [3], [11], [25]. b is well-behaved if the positive portion of 9(b) is closed under the square root operation [19]. A closed ""-derivation is bounded if the positive portion is closed under the square root operation [15]. b is well-behaved if there is a sequence of self-adjoint elements (h ) in 91 such that b(x)=lim i [h n9 x] (x 9(b)) [18]. One can easily see that the (infinitesimal) generator b of a strongly continuous one-parameter group of ""-automorphisms on 91 is always well-behaved and the differentiation djdt on the unit interval is quasi well-behaved. 3. Domain of closed derivations. In mathematical physics, unbounded derivations are often defined by Hamiltonians. In those cases, it is easily seen that the derivations are closable. If 91 is commutative, b is closed and f^c^r) (continuously differentiable), then f(a) 9(b) for a 9(b) s. For a noncommutative 91, one has to replace C\R) by C\R) [6], [14], [17]. 4. Differentiations. Let / be the unit interval and b 0 =d/dt. If b is a derivation in the commutative C ""-algebra C(I) such that 9(b) = C"(I) (77-times continuously differentiable) for some n (n= 9 1,2,3,...), then there is a unique continuous function X on J suchthat b=xb Q. In particular, b is closable [25]. Suppose that b 09 b are two ""-derivations in C(K) (K 9 a compact Hausdorff space), b 0 is closed and 9(b)=9(b Q ) or (XLi^O^o)' ^Gn ^ere * s a un i<l ue continuous function X on K such that b=xb Q. In particular, b is closable [3]. It is an open question whether the result can be extended to n = 2 9 3,.... CONJECTURE. Let b 0 be a closed ""-derivation in 91 and let b be a ""-derivation in 91 with 9(b)=9(b Q ). Then b is closable. It would be an interesting problem to study the relationship between closed ""-derivations in C(K) and differential structures in K. 5. Generators. Let {g(t)} (t R) be a strongly continuous one-parameter group of *-automorphisms on 91 with identity. The system {91, Q (t)} is said to be a C ""-dynamics. It is said to be approximately inner if there is a sequence of uniformly continuous one-parameter groups {# (0} of inner ""-automorphisms on 91 such that \\Q n (t)(a) Q(t)(a)\\-+0 uniformly on every compact subset of R for each fixed # 91. We shall denote this by strong lim g n (t) = g(t). All C""-dynamics appearing in quantum lattice systems and Fermion field theory are approximately inner. In mathematical physics, we are often concerned with a C"-algebra 91 containing an identity and an increasing sequence of C*-subalgebras {9t } of 91 such that lç9t and the uniform closure of Ur=i^n * s ^- * n addition, we are given a

3 Unbounded Derivations in C*-Algebras 711 " -derivation ö in 91 satisfying the following conditions: (1) S(S) = \J^ssl^l n ; (2) there is a sequence of selfadjoint elements (h ) in 91 such that b (a) = i[h U9 a] (fl 9I ). We shall call such a derivation a normal ""-derivation in 91. If a normal "'-derivation b satisfies the following conditions: (1) A 9ï 7l+p for a fixed p; (2) there is an element k n in 9I such that \\h -k \\ = 0(n) 9 then the closure 5 of b is a generator and strong limexprô,.,, =exp tb 9 where b ih (x) = i[h n9 x] (x69t) [4]. (This includes the case of two-dimensional quantum lattice systems with finite range interaction.) If a normal ""-derivation b satisfies \\h n k n \\ = 0(1) for some /c 9I, then b is a generator and strong lim exp tb ih =exp tb [11]. (This includes the case of one-dimensional lattice systems with bounded surface energy.) A normal ""-derivation b is said to be commutative if one can choose the sequence (/? ) such that h n h m =h m h n (m 9 n = l 9 2, 3,...). If b is commutative, then it has an extension b ± suchthat b ± is a generator, strong lim exp tb ih = exp tb l9 and moreover exptb 1 (a)=gxptb ih (a) (tfe9i ) [21]. This result is applicable to all classical lattice systems. There are generalizations of these results to dissipative operators in Banach spaces [3], [4], [9]. 6. Ground states. We introduce a class of states of some importance in quantum physics. Let {9t, g(t)} be a C ""-dynamics and let <5 be the generator of {g(t)}. A state cp on 91 is said to be a ground state for [g(t)} if icp(a*b(a))^q for a 9(b). A ground state is invariant under {g(t)}. If a C"-dynamics is approximately inner, then it has a ground state [18]. There is a nontrivial example of a C "^dynamics without ground state [13]. 7. KMS states. We introduce another class of states on a C* -dynamics which is important in quantum physics. For a real number j?, a state cp ß on 91 is said to be a KMS state for {91, g(t)} at inverse temperature ß if for a, è 91, there is a bounded continuous function F 0)b on the strip 0^1m(z)^ß (0^lm(z)^ß) in the complex plane which is analytic on 0<Im(z)<j3 (0>lm(z)>/?) so that F a,bq)=<pß{aq(t)(b)) and F atb (t+iß) = cp ß (Q(t)(b)a). The KMS condition gives every evidence of being the abstract formulation of the condition for equilibrium of a state [8]. A KMS state is invariant under {g(t)}. If {91, g(t)} is approximately inner and 91 has a tracial state, then it has a KMS state at each inverse temperature ß (-«></}<-foo) [18]. If a C ^dynamics {91, exp tb} has a KMS state cp p at ß and there is a sequence of bounded ^-derivations (b ) on 91 suchthat b (a)-+b(a) for a 9 9 where 9 is a dense subset of 9(b) 9 then {9I,exprô} has a tracial state [10], [12]. There is a C""-dynamics {9I n, Q (t)} which has a KMS state at ß=logn only (n=2 9 3,...) [16]. Let <p ßn be a KMS state at ß n for {91, Q(t)} and ß -+ß' 9 then any accumulation point of {cp ß^ is a KMS state for {9Ï, ç(t)} at ß; if j8 H-oo 5 then any accumulation point of {cp ßi^ is a ground state for {91, g(0}. 8. UHF algebras. A C*-algebra 91 is said to be a UHF algebra if there is an increasing sequence {9Ï } of finite type 7-subfactors such that K9I and the

4 712 Shôichirô Sakai uniform closure of \J7=i^n ^- Such algebras are appearing in quantum lattice systems and Fermion field theory. Let b be a closed ""-derivation in a UHF algebra 91; then there is an increasing sequence {9I } of finite type 7-subfactors in 9(b) suchthat l 9I and U^i 8 *«is dense in 9(b) [5], [21]. We define a normal ""-derivation in a UHF algebra more restrictively than general cases. A ""-derivation b in a UHF algebra 91 is said to be normal if there is an increasing sequence {9I } of finite type 7-subfactors such that l 9l = \J7 =1 Mn- Let ô be a normal ""-derivation and let {e 1! j \i 9 j=l 9 2,... 9 p } be a matrix unit of 9Ï H. Set ^7H=^=i^( e, ji) e ijj the* 1 ö(a)=i[h U9 a] (fl 9I ). It is easily seen that h is a selfadjoint element. All derivations appearing in quantum lattice systems and all quasi free derivations in the canonical anticommutation relation algebra have normal ""-derivations as their cores. Let b be a generator; then there is a normal ""-derivation such that Sczb and 9(h) is contained in the *-subalgebra A(b) of all analytic elements with respect to b [21]. CONJECTURE. Any C""-dynamics {91, g(t)} with a UHF algebra 91 is approximately inner. PROBLEM. Let b be a generator in a UHF algebra. Then can we find a normal ""-derivation 8 suchthat S is the core of b (i.e., the closure of =b)l 9. Bounded perturbations. Let b be a normal ""-derivation in a UHF algebra 91 with 9(b) \J < =1 & n. Then for e>0, there is a normal ""-derivation b E such that 9(b)=9(b E ) 9 b E (@(b))cz9(b) and b-b e is a bounded ""-derivation with II^ <y < 17] > [25]. This implies that an infinite range interaction may move to a finite range interaction by bounded perturbations. Next suppose that <5(^(<5))c 9(b); then by choosing a suitable subsequence of {9Ï }, we may assume that <5(9I )<z9i n+1. Then we have the following decomposition: b=b 1 ~\-b 2t9 where b l9 b 2 are normal ""-derivations with 9(b)=9(b^=9(b^9 <5i(9I 2 )c:9i 2 and <5 2 (9I 2n+1 )c9i In particular, b ± and <5 2 are commutative normal ""-derivations 17]» [25]. Generally, let {91, exp tb} be a C ""-dynamics and let b 0 be a bounded "-derivation on 91; then a C""-dynamics {9Ï, exp t(b+b 0 )} has a ground state (a KMS state at ß) if and only if {91, exp tb} has a ground state (a KMS state at ß) [1]. 10. Phase transition. Let {91, Q(1)} be a C*-dynamics. Suppose that it has a KMS state at every ß (-oo< i 8<- -oo), if it has only one KMS state at /?, then we say that the dynamics has no phase transition at ß. If it has at least two KMS states at ß 9 then we say that it has phase transition at ß. If {91, exp tb} has no phase transition at /?, then {91, exp t(b+b 0 )} has no phase transition at ß for a bounded ""-derivation b 0 [l]. Let b bea normal ^-derivation in a UHF algebra with 9(b) U f 7 =1 9I. Let P n be the canonical conditional expectation of 91 onto 9I such that i(xa) = T(P n (x)a) (flg9t ), where % is the unique tracial state on 91. Let (A n ) be a sequence of self-adjoint elements in 91 such that b(a) = i[h n9 a] (flg9ï ).

5 Unbounded Derivations in C*-Algebras 713 Then, if \\h n P n (h n )\\ = 0(l) 9 then b is a pregenerator and the approximately inner C""-dynamics {9Ï, exp is} has no phase transition at every ß ( «></?<+ «>) [2], [11], [23], [24]. This implies that a quantum lattice system with bounded surface energy has no phase transition at every ß and quasi free C ""-dynamics in the canonical anticommutation relation algebra has no phase transition at every ß. To develop the theory of phase transition for normal ""-derivations more deeply, wc need to find how to construct all KMS states at each ß. For commutative normal ""-derivations, we have a fairly detailed description of the construction of all KMS states at ß. Let b be a commutative normal ""-derivation in a UHF algebra 9Ï such that b(9(b))cz9(b). Then there is a sequence (h ) of self-adjoint elements in Ur =1 9t w such that b(a)=i[h n9 a] (fl69t n ) and h n h m =h m h (m 9 n=l 9 2,3,...). Let fß H be a ""-subalgebra of 9Ï generated by 9I n,a rt. Since 93 c9l m for some m 9 93 isfinite-dimensional.set & : n =2j'Li^n z n,j> where z nj is the minimal central projection of 33. Now let cp ß be a KMS state at ß for the C ""-dynamics {9Ï, g(t)} with Q(t)(a)=oxp t b ih (a) (ö 6 9Ï W ) ; then there is a unique family (h ) of mutually commuting self-adjoint elements in 91 such that h n Ç.SB, b(d) i[ìt n9 b] (& $ ) and q} ß (b)=z(bcxp(-ßh n )) for ft 9ï (n=l, 2,3,...) [24]. This result can be extended to commutative normal ""-derivations with infinite range interactions [24], PROBLEM. Can we find how to construct all KMS states at each ß for normal ""-derivations? References 1. H. Araki, Pubi. Res. Inst. Math. Sci. 9 (1973), Comm. Math. Phys. 44 (1975), C. Batty, preprints. 4. O. Bratteli and A. Kishimoto, preprint. 5. O. Bratteli and D. Robinson, Comm. Math. Phys. 42 (1975), Comm. Math. Phys. 46 (1976), G. Elliott, Invent. Math. 9 (1970), R. Haag, N. Hugenholtz and M. Winnink, Comm. Math. Phys. 5 (1967), P. J0rgensen, J. Functional Anal. 23 (1976), Comm. Math. Phys. 11. A. Kishimoto, Comm. Math. Phys. 47 (1976), P. Kruszynski, Bull. Acad. Polon. Sci. 13. C. Lance and A. Niknam, Proc. Amer. Math. Soc. 61 (1976), A. Mcintosh, preprint. 15. S. Ôta, J. Functional Anal, (to appear). N 16. D. Olsen and G. Pedersen, preprint. 17. R. T. Powers, J. Functional Anal. 18 (1975), R. T. Powers and S. Sakai, Comm. Math. Phys. 39 (1975), J. Functional Anal. 19 (1975), S. Sakai, Tôhoku Math. J. 12 (1960), Amer. J. Math. 98 (1976),

6 714 Shôichirô Sakai; Unbounded Derivations in C*-Algebras 22. Comm. Math, Phys. 43 (1975), J. Functional Anal. 21 (1976), Tôhoku Math. J. 28 (1976), Lecture notes, Univ. of Copenhagen and Newcastle upon Tyne, G. Silov, Dokl. Akad. Nauk. S.S.S.R. 58 (1947). UNIVERSITY OF PENNSYLVANIA PHILADELPHIA, PENNSYLVANIA 19174, U.S.A.