An element a of norm one in a JB*-triple A is said to be smooth if there exists a unique element x in the unit ball A1* of the dual A* of A at which a attains its norm, and is said to be Fréchet-smooth if, in addition, any sequence (xn) of elements in A1* for which (xn(a)) converges to one necessarily converges in norm to x. The sequence (a2n+1) of odd powers of a converges in the weak*-topology to a tripotent u(a) in the JBW*-envelope A** of A. It is shown that a is smooth if and only if u(a) is a minimal tripotent in A** and a is Fréchet-smooth if and only if, in addition, u(a) lies in 

Edwards, C. Martin

Rüttimann, Gottfried T.

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SMOOTHNESS PROPERTIES OF THE UNIT BALL IN AJB*-TRIPLEC. MARTIN EDWARDS AND GOTTFRIED T. RUTTIMANNABSTRACTAn element a of norm one in a JB*-triple A is said to be smooth if there exists a unique element xin the unit ball A* of the dual A* of A at which a attains its norm, and is said to be Frechet-smooth if, inaddition, any sequence (xn) of elements in A* for which (xn(a)) converges to one necessarily converges innorm to x. The sequence (a2n+1) of odd powers of a converges in the weak*-topology to a tripotent u(a)in the JBW*-envelope A** of A. It is shown that a is smooth if and only if u(a) is a minimal tripotent inA** and a is Frechet-smooth if and only if, in addition, u{a) lies in A.1. IntroductionOver several years, the authors have studied the facial structure of the unit ballsin a Banach space and its dual [5, 6, 7, 8, 9, 10]. This note is concerned with twosmoothness properties of the unit ball in a complex Banach space. In a recent paper[19], Taylor and Werner studied these properties for the unit ball of a C*-algebra andwere able to give several algebraic conditions which were equivalent to smoothness.The purpose of this paper is to consider the same properties for the unit ball in a JB*-triple. The main result shows that both smoothness conditions are equivalent toalgebraic conditions which, necessarily, are expressed in terms of the triple product.This leads to further equivalent new conditions for the case of a C*-algebra.2. PreliminariesLet V be a complex vector space, and let C be a convex subset of V. A convexsubset E of C is said to be a face of C provided that, if tal + (\ —t)a2 is an elementof E, where ay and a2 lie in C and 0 < / < 1, then ax and a2 lie in E. A face E of C issaid to be proper if it differs from C. An element a in C for which {a} is a face is saidto be an extreme point of C. Let T be a locally convex Hausdorff topology on V, andlet C be r-closed. Let ^;(C) denote the set of r-closed faces of C. Both 0 and C areelements of «^(C), and the intersection of an arbitrary family of elements of &\{C)again lies in &\(C). Hence, with respect to ordering by set inclusion, &\{C) forms acomplete lattice. A subset E of C is said to be a z-exposed face of C if there exists ar-continuous linear functional / o n Fand a real number t such that, for all elementsa in C\E, Ref{a) < t and, for all elements a in E, Re/(a) = /. Let $X(C) denote the setof r-exposed faces of C. Clearly, SX{C) is contained in i^(C), and the intersection ofa finite number of elements of SX{C) again lies in $X(C). The intersection of anarbitrary family of elements of $Z(C) is said to be a r-semi-exposedface of C. Let «^(C)denote the set of i-semi-exposed faces of C Clearly, < .^(C) is contained in £fr(C), andReceived 11 April 1994.1991 Mathematics Subject Classification 46L10.Research supported in part by a grant from the Schweizerische Nationalfonds/Fonds national suisse.Bull. London Math. Soc. 28 (1996) 156-160SMOOTHNESS PROPERTIES OF THE UNIT BALL 157the intersection of an arbitrary family of elements of «^(C) again lies in «^(C). Hence,with respect to ordering by set inclusion, 5^(C) forms a complete lattice and theinfimum of a family of elements of 5^(C) coincides with its infimum when taken inWhen V is a complex Banach space with dual space V*, the abbreviations n andw* will be used for the norm topology of V and the weak*-topology of V*. For eachsubset E of the unit ball Vx in V and F of the unit ball V* in V*, let the subsets E'and F, be defined byE' = {xe V*: x(a) = 1 for all aeE}, F, = {ae Vx: x(a) = 1 for all xeF}.Notice that E lies in &>n(Vx) if and only if {E'), = E, F lies in «9^(F*) if and only if(F,)' = F, and the mappings E-+E' and F-> F, are anti-order isomorphisms betweenSfn{Vx) and ^(Vx) and are inverses of each other. The reader is referred to [8] fordetails.An element a in Vx is said to be smooth if {a}' coincides with {x} for some x in V*.It follows that the extreme point x of V* is weak*-exposed. A smooth point a is saidto be Fre'chet-smooth if, for each sequence (xn) in V* such that the sequence (xn(a))converges to one, it follows that (jcn) converges in norm to x. For the equivalence ofthese definitions with the conventional ones, the reader is referred to [18] and [19].LEMMA 2.1. Let V be a complex Banach space, let W be a closed subspace of V,and let a be an element of norm one in W. If a is smooth in V, then a is smooth in W,and if a is Fre'chet-smooth in V, then a is Fre'chet-smooth in W.Proof. Suppose that the element a in W of norm one is smooth in V. Then thereexists a unique element JC of norm one in V* such that x{a) = 1. The restriction x0 ofx to W has the property that xo(a) = 1, and is therefore of norm one. Let y be anelement of W* of norm one such that y{a) = 1. Then, by the Hahn-Banach theorem,y can be extended to an element z of V* of norm one. Since a is smooth, z and xcoincide, and therefore y is the restriction of x to W. It follows that a is smooth inW. Now suppose that a is Frechet-smooth in V, and suppose that (yn) is a sequenceof elements of W\ such that (yn(a)) converges to one. By the Hahn-Banach theorem,there is a sequence (:cn) in V* such that yn is the restriction of xn to W. The sequence(xn(a)) converges to one and, by the Frechet-smoothness of a in V, it follows that (xn)converges in norm to x. However,and therefore (yn) converges in norm to x0 as required.The following result is well known, but a proof will be given for completeness.LEMMA 2.2. Let Qbe a locally compact Hausdorjf space, and let C0(Q) denote thecomplex Banach space of continuous complex-valued functions on Q. vanishing atinfinity, endowed with the supremum norm. Let a be a positive element ofC0(Q) of normone. Then a is smooth if and only if there exists a unique element co0 in Q such thata(co0) = 1, and a is Fre'chet-smooth if and only if, in addition, co0 is an isolated point.Proof. The dual of C0(Q) may be identified with the space M(Q) of complexregular Borel measures of finite total variation on Q. Let a be a positive smooth point158 C. MARTIN EDWARDS AND GOTTFRIED T. RUTTIMANNin the unit ball in C0(Q). There exists a point co0 in Q such that a(co0) = 1. Since a issmooth, it follows that the unit point measure Sm is the unique element of M(Q) suchthat<5Wo(0)=l.Suppose that a is Frechet-smooth and co0 is not isolated. Then, for each positiveinteger n, the open set {co: \a(co) — \\ < \/n) contains a point con not equal to coQ. Thesequence (dw ) of unit point measures is such that the sequence (Sm (a)) converges toone. However, for each n there exists a positive element b of C0(Q) such that8m (b) = 1 and da (b) = 0. Hence (3a ) does not converge in norm to 5m . This yieldsa contradiction.Recall that a complex vector space A equipped with a triple product(a,b,c) ->{abc} from Ax Ax A to A which is symmetric and linear in the first andthird variables, conjugate linear in the second variable and satisfies the identity[D(a, b), D(c, d)] = D({a b c), d) - D(c, {da b}),where [, ] denotes the commutator and D is the mapping from Ax A to A defined byD(a,b)c = {abc},is said to be a Jordan*-triple. When A is also a Banach space such that D is continuousfrom Ax A to the Banach space B(A) of bounded linear operators on A, and, for eachelement a in A, D(a,a) is hermitian with non-negative spectrum and satisfies\\D{a,a)\\ = \\a\\\then A is said to be a JB*-triple. A JB*-triple which is the dual of a Banach space issaid to be a JBW*-triple.Examples of JB*-triples are C*-algebras, and examples of JBW*-triples areW*-algebras. The second dual A** of a JB*-triple A possesses a triple product withrespect to which it is a JBW*-triple, the canonical mapping from A into A** being anisomorphism. For details, the reader is referred to [1, 2, 3, 4, 11, 12, 13, 14, 15, 16,17, 20].An element u in a JBW*-triple A is said to be a tripotent if {u u u) is equal to u. Theset of tripotents in A is denoted by <%(A). A pair u, v of elements of <%(A) is said tobe orthogonal if {v v u} = 0. For two elements u and v of %(A), write u ^ v if {u v u) = uor, equivalently, if v - u is a tripotent orthogonal to u. This defines a partial orderingon %(A) with respect to which %{A) with a greatest element adjoined forms acomplete lattice denoted by %{A)~. When A is a W*-algebra, the set of tripotents inA coincides with the set of partial isometries in A.The proof of the following result can be found in [8].LEMMA 2.3. Let A be a JBW*-triple with predual A+ and let °U{AJ be the completelattice of tripotents in A with a largest element adjoined.(i) Every norm-closed face of the unit ball Atl in A+ is norm-exposed and of theform {u}, for some u in °U{Af.(ii) Every weak*-closed face of the unit ball Al in A is weak*-semi-exposed and ofthe form ({«},)' for some u in <%{A)~.(iii) The map u\-*{u}, is an order isomorphism from the complete lattice °U(A)~ ontothe complete lattice ^(A^), and the map MI—• ({«},)' is an anti-order isomorphismfrom %{A)" onto the complete lattice ^(AJ.SMOOTHNESS PROPERTIES OF THE UNIT BALL 159For each element a in a JBW*-triple A, the odd powers a2n+l are definedinductively bya1 = a, a2n+1 = {aa2n~1a}.The proof of the following lemma can be found in [8].LEMMA 2.4. Let a be an element of norm one in a JBW*-triple A. Then thesequence (a2n+1) converges in the weak*-topology to a tripotent u(a) such that the norm-closed faces {a}, and {u(a)}, of the unit ball A^ of the predual A+ of A coincide.3. Main resultsThe main result of the paper can now be given using those above. Let A bea JB*-triple that will be regarded as being canonically embedded in its JBW*-envelope A**.THEOREM 3.1. Let A be a JB*-triple, and let a be an element of norm one in A.(i) The point a is smooth if and only ifu(a) is a minimal non-zero element of thecomplete lattice <%(A**)~.(ii) The point a is Frechet-smooth if and only ifu(a) is a minimal non-zero elementof the complete lattice ^1{A**)" and is contained in A.Proof. Suppose that a is smooth, in which case {a}' consists of the set [x], forsome extreme point JC of A*. By [11, Proposition 4] and Lemma 2.4, it follows thatu(a) is a minimal non-zero element of °U(A**y. Conversely, if u{a) is non-zero andminimal, again using [11, Proposition 4], there exists a unique extreme point x of A*such that{a}' = {u(a)}, = {x}and a is smooth.Let C(a) be the JB*-subtriple of A generated by the odd powers of a. By [14,Lemma 1.14], C(a) is isomorphic to the commutative C*-algebra C0(Q) of continuousfunctions on a locally compact Hausdorff space Q which vanish at infinity, andunder this isomorphism a is mapped into a positive element. That a is Frechet-smoothif and only if u(a) is contained in A follows immediately from Lemmas 2.1 and 2.2.When A is a C*-algebra, the partial isometry u{a) corresponding to an element ain A of norm one is the weak*-limit of the sequence (a2n+1), wherea1 = a, a2n+1 = aa*(a2n~l).Theorem 3.1 leads to the following result for C*-algebras.COROLLARY 3.2. Let a be an element of norm one in the C*-algebra A.(i) The point a is smooth if and only if the partial isometry u(a) is such thatu(a)u(a)*A**u(a)*u(a) = Cu(a).(ii) The point a is Frechet-smooth if and only if a is smooth and u(a) is containedin A.For several equivalent conditions in the case of a C*-algebra, the reader is referredto [19].160 C. MARTIN EDWARDS AND GOTTFRIED T. RUTTIMANNReferences1. T. J. BARTON, T. DANG and G. HORN, 'Normal representations of Banach Jordan triple systems',Proc. Amer. Math. Soc. 102 (1987) 551-555.2. T. J. BARTON and R. M. TIMONEY, 'Weak* continuity of Jordan triple products and its applications',Math. Scand. 59 (1986) 177-191.3. S. DrNEEN, 'Complete holomorphic vector fields in the second dual of a Banach space', Math. Scand.59(1986) 131-142.4. S. DINEEN, 'The second dual of a JB*-triple system', Complex analysis, functional analysis andapproximation theory (North-Holland, Amsterdam, 1986).5. C. M. EDWARDS and G. T. RUTTIMANN, 'Isometries of GL-spaces', J. London Math. Soc. 31 (1985)125-130.6. C. M. EDWARDS and G. T. RUTTIMANN, 'On the facial structure of the unit balls in a GL-space andits dual', Math. Proc. Cambridge Philos. Soc. 98 (1985) 305-322.7. C. M. EDWARDS and G. T. RUTTIMANN, 'On the facial structure of the unit ball in a GM-space',Math. Z. 193 (1986) 597-611.8. C. M. EDWARDS and G. T. RUTTIMANN, 'On the facial structure of the unit balls in a JBWMriple andits predual', J. London Math. Soc. 38 (1988) 317-322.9. C. M. EDWARDS and G. T. RUTTIMANN, 'A characterization of inner ideals in JB*-triples', Proc.Amer. Math. Soc. 116 (1992) 1049-1057.10. C. M. EDWARDS and G. T. RUTTIMANN, 'Structural projections on JBW*-triples\ J. London Math.Soc, to appear.11. Y. FRIEDMAN and B. Russo, 'Structure of the predual of a JBW*-triple', J. Reine Angew. Math. 356(1985) 67-89.12. Y. FRIEDMAN and B. Russo, 'The Gelfand-Naimark theorem for JBMriples', Duke Math. J. 53(1986) 139-148.13. N. JACOBSON, Structure and representation of Jordan algebras, Amer. Math. Soc. Colloq. Publ. 39(Amer. Math. Soc, Providence, RI, 1968).14. W. K.AUP, 'Riemann mapping theorem for bounded symmetric domains in complex Banach spaces',Math. Z. 183 (1983) 503-529.15. O. Loos, Jordan pairs, Lecture Notes in Math. 460 (Springer, Berlin, 1975).16. K. MEYBERG, 'Jordan-Tripelsysteme und die Koecher-Konstruktion von Lie-Algebren', Math. Z. 115(1970) 58-78.17. E. NEHER, Jordan triple systems by the grid approach, Lecture Notes in Math. 1280 (Springer, Berlin,1987).18. K. F. TAYLOR and W. WERNER, 'Differentiability of the norm in von Neumann algebras', Proc. Amer.Math. Soc. 119 (1993) 475-480.19. K. F. TAYLOR and W. WERNER, 'Differentiability of the norm in C*-algebras', Functional analysis(Essen 1991), Lecture Notes in Pure and Appl. Math. 150 (Dekker, New York, 1994) 329-344.20. H. UPMEIER, Symmetric Banach manifolds and Jordan C*-algebras (North-Holland, Amsterdam,1985).The Queen's College University of BerneOxford OX1 4AW BerneSwitzerland

Smoothness Properties of the Unit Ball in a JB*-Triple

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