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International Journal of Theoretical Physics, Vol 39, No 3, 2000 Lp-Spaces for C*-Algebras with a State† Stanisław Goldstein1 and Phan Viet Thu2 Received December 8, 1999 We present here a construction of noncommutative Lp-spaces for a C*-algebra with respect to a state on the algebra Their properties are deduced from wellestablished properties of corresponding Haagerup and Kosaki spaces Two examples are considered INTRODUCTION There have been several attempts at the construction of noncommutative Lp-spaces for specific C*-algebras (Majewski and Zegarlin´ski, 1995, 1996; Goldstein and Phan, 1998) We give here a general definition and the basic properties of the spaces Further details may be found in Phan (1999) Let M be a von Neumann algebra acting in a Hilbert space H and a normal faithful semifinite weight on M Let {t}t ޒdenote the modular automorphism group on M associated with Recall that the crossed product ލϭ M ϫԽt ޒis a von Neumann algebra acting on ވϭ L2(ޒ, H ) generated by operators M(a), a M and M(s), s ޒ, defined by (M(a))(t) ϭ Ϫt (a)(t) (M(s))(t) ϭ (t Ϫ s) ވ, t ޒ ˜ the algebra of all Let denote the canonical trace on ލ Denote by ލ -measurable operators affliated with ލ The dual representation (or the dual action of ޒon M ϫԽ )ޒis the continuous automorphism representation s ۋs of ޒwhere s is the *automorphism of M ϫԽ ޒthat is implemented by the unitary operator s defined by (s)(t) ϭ eϪits(t), L2 (ޒ, H ), t ޒ For each p [1, ϱ], the Haagerup Lp-space is defined by † This paper is dedicated to the memory of Prof Gottfried T Ruăttimann Faculty of Mathematics, University of Ło´dz´, Ło´dz´, Poland E-mail address: goldstei@math.uni.lodz.pl Faculty of Mathematics, Mechanics and Computer Science, Hanoi National University, Hanoi, Vietnam 687 0020-7748/00/0300-0687$18.00/0 ᭧ 2000 Plenum Publishing Corporation 688 Goldstein and Thu ˜ : ∀s ޒ, sh ϭ eϪs/p h} Lp (M ) :ϭ {h ލ We identify Lϱ (M ) with M by means of M Lemma 1.1 Let M be a von Neumann algebra, a faithful normal state on M, t the modular automorphism group of M associated with , ލϭ M ϫԽ ޒ, the dual weight, the canonical normal faithful semifinite trace on ލ, h ϭ d/d Then, for p [1, ϱ], the mapping ip M → Lp (M ) defined by a ۋip(a) ϭ h1/2p и a и is linear and injective; h1/2p и M0 и h1/2p is if M0 is -weakly dense in M, then ip(M0) ϭ h1/2p p norm-dense in L (M ) for p Ͻ ϱ (-weakly dense for p ϭ ϱ) We have 1 Յ r Յ p Յ ϱ ⇒ ሻir(x)ሻr Յ ሻip(x)ሻp , x M The first part can be found, for example, in Goldstein and Lindsay, 1995 As for the second part of the lemma, it is easy to prove for p ϭ using duality The proof for any p [1, ϱ] can be deduced from the Kaplansky density theorem and the inequality 1/q ሻh1/2p и a и h1/2p ሻp Յ ሻh1/2 и a и h1/2ሻ1/p ሻaሻϱ with 1/p ϩ 1/q ϭ 1, to be found in Terp (1982) or Goldstein and Lindsay, 1999 The last part follows from Hoălders inequality since ir(x) ϭ h1/2s ip(x)h1/2s where s [1, ϱ] satisfies 1/p ϩ 1/s ϭ 1/r DEFINITION OF Lp(A, ) SPACES Let A be a C*-algebra and a state on A; let (H, , ) denote the GNS representation of A associated with In this section we introduce the spaces Lp(A, ) Their properties will be given in the next section First let us specify some notations that we shall use in the sequel is the vector state on (ނH) given by (a) ϭ (a, ), a (ނH) s is the support of the state Ȋ(A)Љ on the von Neumann algebra (A)Љ H denotes the Hilbert space sH (with the inner product inherited from H) M is the von Neumann algebra s(A)Љs acting on H Lp-Spaces for C*-Algebras With a State 689 denotes the faithful normal state ȊM t is the modular automorphism group of M relative to ލis the crossed product M ϫԽ t ޒacting on the Hilbert space ވ ϭ L (ޒ, H ); the image M(M ) of M in ލwill be denoted by M, too ˜ - the s denotes the dual action of ޒon ލor its extension to ލ topological *algebra of -measurable operators affiliated with ލ is the canonical normal faithful semifinite trace on ލ 10 Lp(M ) with p [1, ϱ] is the Haagerup space (consisting of measurable operators affiliated with )ލwith norm ሻ и ሻp 11 h is the measurable operator affiliated with ލdefined by h ϭ d/d, where is the dual weight of Consider the map ␥: A → M given by a ۋs(a)s This is a positive linear contraction with -weakly dense range Let N be the kernel of ␥ and let ␥˜ denote the induced map A/N → M Then N is a closed involutive subspace of A, and the quotient space A/N is a Banach space in the quotient norm, with positive elements of the form [a], for a A+ The injective linear map ␥p :ϭ ip ˜␥ ؠ: A/N → Lp(M ) is positivity preserving, and Lemma 1.1 implies that it has norm-dense range for p Ͻ ϱ and -weakly dense range for p ϭ ϱ Norms are defined on A/N by ሻ[a]ሻp ϭ ሻip(␥(a))ሻLp(M), the resulting normed space is denoted Lp0(A, ) Thus, for p Ͻ ϱ, (Lp(M ), ␥p) is a completion of Lp0(A, ) in which the dense isometric embedding ␥p respects positivity In order to obtain compatible spaces we consider a different family of completions Let (L1(A, ), ) be any completion of L10(A, ) By Lemma 2.1, the norms on A/N satisfy ሻ[a]ሻr Յ ሻ[a]ሻp for Յ r Յ p Յ ϱ Therefore completions (Lp(A, ), p) of Lp0(A, ) may be found satisfying (A/N) ʚ Lp(A, ) ʚ Lr(A, ) ʚ L1(A, ) for Յ r Յ p Ͻ ϱ The positive elements of Lp(A, ), p Ͻ ϱ, are given by Lpϩ (A, ) ϭ closure in Lp(A, ) of p((A/N)+) We denote by ⌫p the unique isometric isomorphism from Lp(A, ) to L (M ) extending the maps ␥˜ and ␥p It is clearly positivity preserving p 690 Goldstein and Thu ϱ Note that the mappings ⌫Ϫ1 p ؠip , p Ͻ ϱ, not depend on p Define L (A, ) to be the image of M under the mappings When restricted to Lϱ ϱ (A, ), the mapping iϪ1 p ⌫ ؠp is an isometric isomorphism from L (A, ) to ϱ L (M ) which extends ␥˜ and ␥ϱ We denote it by ⌫ϱ It follows that the Lp-spaces over a C*-algebra with respect to a state inherit all the standard properties of duality, reflexivity and uniform convexity, and the Hoălder and Clarkson inequalities, from the Haagerup Lp-spaces Note also that Lϱ(A, ) and L1(A, ) form a compatible pair of Banach spaces In the next section we shall fix the multiplicative structure of the spaces and state some of the properties that relate to the structure We shall also show that these Lp-spaces are complex interpolation spaces, by relating them to Kosaki’s Lp-spaces THE PROPERTIES OF Lp(A, ) SPACES Let p, q, r [1, ϱ] be such that 1/p ϩ 1/q ϭ 1/r For a Lp(A, ), b Lq(A, ) define a pиq b Lr(A, ) by a pиq b :ϭ r (p(a) q(b)) Proposition 3.1 (Hoălders inequality) Let r, p, q [1, ϱ] be such that 1/p ϩ 1/q ϭ 1/r, a Lp(A, ), b Lq(A, ) Then ሻa pиq bሻr Յ ሻaሻpሻbሻq • We define a linear functional tr on L1(A, ) by • tr(a) ϭ tr(␥1(a)), a L1(A, ), where tr is the usual linear functional tr on L1(M ) For p, q [1, ϱ], 1/p ϩ 1/q ϭ 1, a, b A/N define • ͗aȊb͘ :ϭ tr(a pиq b) Proposition 3.2 Let p, q [1, ϱ], 1/p ϩ 1/q ϭ It follows that • • tr(a pиq b) ϭ tr(b qиp a); ͗aȊb͘ is independent of p, q [1, ϱ] such that 1/p ϩ 1/q ϭ 1; ͗иȊи͘ is bilinear Proposition 3.3 Suppose that p, q [1, ϱ], 1/p ϩ 1/q ϭ and a Lp(A, ); then • ሻaሻp ϭ sup{Ȋtr(a pиq b)Ȋ: b Lq(A, ), ሻbሻq Յ 1} Proposition 3.4 Let p ]1, ϱ] and 1/p ϩ 1/q ϭ Lp-Spaces for C*-Algebras With a State 691 Let a Lp(A, ); then a defined by • a(b) :ϭ tr(a pиq b), b Lq(A, ), is a bounded linear functional on Lq(A, ) The mapping a ۋa is an isometric isomorphism of Lp(A, ) onto the dual Banach space of Lq(A, ) Proposition 3.5 (L2(A, ), ሻ и ሻ2) is a Hilbert space with the inner product • • (aȊb)L2(A,) :ϭ tr(b* 2и2 a)(ϭ tr(a 2и2 b*)) for a, b L2(A, ) We turn now to interpolation Let Lp(M ) denote the Kosaki spaces defined by Lp (M ) ϭ h1/2q и Lp (M ) и h1/2q ʚ L1 (M ), p, q [1, ϱ], 1/p ϩ 1/q ϭ 1, with the norm p ሻh1/2q и x и h1/2q ሻp ϭ ሻxሻp for x L (M ) We know that Lϱ (A, ) ʚ Lp (A, ) ʚ Lq (A, ) ʚ L1 (A, ) for q [1, ϱ], q Յ p and that Lϱ (A, ) and L1 (A, ) form a pair of compatible Banach spaces Denote by C(X0, X1) the Calderon’s complex interpolation functor for the pair of compatible Banach spaces (X0, X1) (Berg and Loăfstroăm, 1976; Calderon, 1964; Kosaki, 1984) We refer now to the paper of Kosaki, 1984 Using his notation from sections 8, 9, we put 0 ϭ 0 ϭ , h0 ϭ k0 ϭ h and ϭ 1/2 where , h are our normal faithful state on M and the corresponding Radon-Nikodym derivative defined at the beginning of this section We consider now the isometry 1/2 ⌫1 : L1 (A, ) → L1 (M ), ⌫1ȊA/N : a ۋh1/2 и ⌫ϱ (a) и h Then the restriction of ⌫1 to Lp(A, ) is an embedding of Lp(A, ) into the Haagerup L1(M ) space such that ⌫1(Lp(A, )) is exactly the Kosaki complex interpolation space C1/p(M1/2, M*) In fact, it is clear that ⌫1 is an isometric isomorphisms from L1(A, ) to L1(M ) We have ϱ 1/2 ⌫1(Lϱ(A, )) ϭ h1/2 и ⌫ϱ(L (A, )) и h ϭ i1(Lϱ(M )) ϭ Lϱ(M ) We easily check that ⌫1 restricted to Lϱ(A, ) takes the space isometrically 692 Goldstein and Thu onto Lϱ(M ) For p ]1, ϱ[, let q ]1, ϱ[ be s.t 1/p ϩ 1/q ϭ 1; then, for a A/N, и ␥p (a) и h1/2q ␥1 (a) ϭ h1/2q Thus ⌫1 (Lp(A, )) ϭ h1/2q и Lp(M ) и h1/2q ϭ Lp(M ) It is routine to check that ⌫p is an isometric isomorphism from Lp(A, ) to Lp(M ) We conclude the following Theorem 3.6 C1/p(L1(A, ), Lϱ(A, )) ϭ Lp(A, ), that is our Lp-spaces are interpolation spaces EXAMPLES In view of possible applications (Majewski and Zegarlin´ski, 1995, 1996), it is important to know how the Lp-spaces behave under inductive limits We exhibit two situations in which they behave well Theorem 4.1 Let (A, ␣t) be a C*-dynamical system and {Aj}jI a generating nest of C*-subalgebras of A, invariant under {␣t} Let be an ␣t-KMS state on A Then, for p [1, ϱ[, Lp(A, ) is an inductive limit of {Lp(Aj , j)}jI where j ϭ ȊAj , j I and, moreover, Lp(A, ) Х Lp((A)Љ) Theorem 4.2 Let A be a UHF C*-algebra with a generating nest {An} and a product state on A with respect to the sequence {An} Suppose that for each i, i is faithful Then for p [1, ϱ], Lp(A, ) is the inductive limit of {Lp (An , (n))}; moreover, Lp (A, ) Х Lp ( (A)Љ) ACKNOWLEDGEMENT Full credit for the improvements in this revised version of the paper goes to J Martin Lindsay The first named author was supported by KBN grant 2P03A 044 10 REFERENCES Berg, J and Loăfstroăm J (1976) Interpolation spaces: An introduction, Springer Verlag, Berlin Calderon, A P (1964) Intermediate spaces and interpolation, the complex method, Studia Mathematica, 24, 113–190 Goldstein, S and Lindsay, J M (1995) KMS symmetric Markov semigroups, Mathematische Zeitschrift, 219, 591–608 Lp-Spaces for C*-Algebras With a State 693 Goldstein, S and Lindsay, J M (1999) Noncommutative interpolation and Markov semigroups, preprint Goldstein, S and Phan V T (1998) Lp-spaces for UHF algebras, International Journal of Theoretical Physics, 37, 593–598 Kosaki, H (1984) Application of the complex interpolation method to a von Neumann algebra: Non-commutative Lp-spaces, Journal of Functional Analysis, 56, 29–78 Majewski, A W and Zegarlin´ski, B (1995) Quantum stochastic dynamics I: Spin systems on a lattice, Mathematical Physics Electronic Journal (2) Majewski, A W and Zegarlin´ski, B (1996) On quantum stochastic dynamics and non-commutative Lp-spaces, Letters in Mathematical Physics, 36, 337–349 Phan V T (1999) Lp-spaces for C*-algebras, Doctoral Thesis, University of Lo´dz´ Terp, M (1982) Interpolation spaces between a von Neumann algebra and its predual, Journal of Operator Theory 8, 327–360 ... from the Haagerup Lp-spaces Note also that Lϱ (A, ) and L1 (A, ) form a compatible pair of Banach spaces In the next section we shall fix the multiplicative structure of the spaces and state some... Functional Analysis, 56, 29–78 Majewski, A W and Zegarlin´ski, B (1995) Quantum stochastic dynamics I: Spin systems on a lattice, Mathematical Physics Electronic Journal (2) Majewski, A W and... and Zegarlin´ski, B (1996) On quantum stochastic dynamics and non-commutative Lp-spaces, Letters in Mathematical Physics, 36, 337–349 Phan V T (1999) Lp-spaces for C* -algebras, Doctoral Thesis,
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