THE MULTIPLICATIVITY OF THE MINIMAL INDEX OF SIMPLE C'-ALGEBRAS

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1 proceedigs of the america mathematical society Volume 123, Number 9, September 1995 THE MULTIPLICATIVITY OF THE MINIMAL INDEX OF SIMPLE C'-ALGEBRAS SATOSHI KAWAKAMI AND YASUO WATATANI (Commuicated by Palle E. T. Jorgese) Abstract. We show the multiplicativity of the miimal idex for simple C- algebras. Although our proof is very short ad elemetary, it is also valid for subfactors, which was first show by Kosaki ad Logo (1992). 1. INTRODUCTION The multiplicativity of Joes idex [9] for subfactors of type II i-factors is a basic fact: If N c L c M are factors of type II i, the [M : N] = [M : L][L : N]. For a iclusio of ifiite factors N c M, Idex E was itroduced by Kosaki [12], depedig o a ormal faithful coditioal expectatio E of M oto N. Amog such coditioal expectatios, oe ca choose a uique coditioal expectatio E0 such that Idex E0 < Idex E for ay E : M» N as i Havet [5], Hiai [6] ad Logo [14, 15]. This P0 is called a miimal coditioal expectatio, ad Idex P0 is called the miimal idex of N c M, ofte deoted by [M : N]o. The multiplicativity of the miimal idex was show by Kosaki-Logo [13] i the case of iclusios obtaied by basic costructios, reducig to the result i Pimser-Popa [19, 20]. The geeral case for subfactors was proved by Logo [16] i his sector theory, applyig the above result [13]. H. Kosaki has also iformed us that R. Logo had a direct proof free from sector theory [17]. Popa [21] also gives a alterative proof for type IIrfactors. Yamagami [23], Deizeau ad Havet [3] have other approaches. Moreover the first-amed author has also cosidered it i a geeral situatio ad show that it is related to the chai rules of idicial derivatives for vo Neuma subalgebras [11]. Owig to [6, 7] ad [10, 11], the multiplicativity of the miimal idex is kow to be closely related to the additivity of the relative etropy of Coes-Stormer [2] ad Pimser-Popa [19]. See [4, 18] also for the basic otios of subfactors. I this ote, we shall show the multiplicativity of the miimal idex for simple C* -algebras. Although our proof is very short ad elemetary, it is also Received by the editors February 14, Mathematics Subject Classificatio. Primary 46L37; Secodary 46L America Mathematical Society 2809 Licese or copyright restrictios may apply to redistributio; see

2 2810 SATOSHI KAWAKAMI AND YASUO WATATANI valid for subfactors. We do ot require ay kowledge o sectors, etropy, or the Takesaki duality theorem. 2. Idex for C*-subalgebras We recall some otatios ad properties o the idex for C*-subalgebras from [22]. Let B be a uital C*-algebra ad A a C*-subalgebra with the same uit /. Let E be a coditioal expectatio of B oto A. Throughout this ote, coditioal expectatios are assumed to be faithful. The, E is called of idex-fiite type if there exists a fiite set {ux, u2,..., u } c B, called a basis for E, such that x = y^ UiE(u*x) for ay x B. (A fiite family {(ux, u\), (u2, u2),..., (u, u*)} is a quasi-basis for E i the sese of [22].) Whe E is of idex-fiite type, the idex of E is defied by Idex E = ^TujU*. 1=1 The value Idex E does ot deped o the choice of a basis for E, ad Idex E is i Ceter B, the ceter of B. See Izumi [8] for iterestig examples of simple C*-subalgebras of Cutz algebras with fiite idex. Whe A c B is a factor-subfactor pair, Idex E coicides with Kosaki's idex [12]. Let q be a actio of a fiite group G o a C*-algebra A ad B = A xag the crossed product. The, there is a caoical coditioal expectatio E of B oto A such that E('S~) agxg) = aei for \^ agxg A x G = B. geg geg I this situatio, {Xg} is a basis for E. If a operator x commutes with A, the J2g G -gxa-g obviously commutes with B because B is geerated by A ad {Xg\g G}. This fact suggests the followig lemma: Lemma 1. Let A c B c C be iclusios of uital C*-algebras with the same uit ad E : B» A a coditioal expectatio of idex-fiite type with a basis {ux, u2,...,u} for E. The, for x A' C, Y! =x u xu* is i B' C. Proof. For x A' D C, "=, u xu* commutes with ay b B. Ideed, b(^2ujxu*) = ~^2(bu,)xu* y y UjE(u*bu )xu* 1=1 '=1 1=1 v'=l = 5Z 5Z ujxe(u*buj)u* (sice E(u*bu,) A ) 7 = 1 1=1 = Y^ UJXC^2 E(u*jbuj)u*) = Y2 UjX(u*b) = C^2 u xu*)b 7 = 1 í=l 7=1 1=1 Therefore, we have Yf =x u,xu* B' C. Licese or copyright restrictios may apply to redistributio; see

3 THE MINIMAL INDEX OF SIMPLE C-ALGEBRAS 2811 Remark A. (1) I the case C = B, Lemma 1 asserts that ] "_, u xu* Ceter B for x A' B, especially, Idex E = ^ «,«,* e Ceter P. (2) I the case C = B(H) for some Hubert space H, Lemma 1 suggests that «F(x) = ^2 u xu* (x A') defies a ormal bouded operator valued weight F of A' oto B'. Ideed, whe Ac B is a factor-subfactor pair, it is kow that F = E~x by [22], [7]. 3. Miimal idex We recall that oe ca miimize idices of coditioal expectatios of uital C*-algebras with trivial ceters, for example uital simple C*-algebras. Propositio 2 [22]. Let A c B be a iclusio of uital C*-algebras with Ceter A = Ceter B = C/. Assume that there exists a coditioal expectatio F : B A of idex-fiite type. The, there exists a uique miimal coditioal expectatio Eo : B -* A, i.e., Idex Eo < Idex E for ay coditioal expectatio E : B» A. Moreover, E = Eo if ad oly if y^ u xu* = ce(x) (x Ä B) for some costat c > 0, where {ux, u2,..., u} is a basis for E. Remark B. ( 1 )The above costat c is give by c = Idex E. (2) Idex P0 is called the miimal idex for a pair A c B of C*-algebras ad is ofte deoted by [B : A]o. (3) Whe A c B is a factor-subfactor pair of fiite idex, every coditioal expectatio of B oto A is automatically ormal ad of fiite idex. Therefore, observig Remark A(2), we see that the above Propositio 2 is exactly the same as Hiai's characterizatio of miimal idex i [6]. Now we are ready to describe the mai theorem, which asserts the multiplicativity of the miimal idex for uital simple C*-algebras. Theorem 3. Let A c B c C be iclusios of uital C*-algebras with Ceter A = Ceter B = Ceter C = CI. Let E :B -» A ad F : C -» B be coditioal expectatios of idex-fiite type. The, Pop is miimal if ad oly if E ad F are miimal. Moreover, miimal idex is multiplicative, that is, [C : A]o = [C : B]0 [B : A]0 Proof. Suppose that E o F is miimal. The, the fact that Idex(Pop) = (Idex P)(Idex F) [22, Propositio 1.7.1] implies that both E ad P eed to be miimal. Coversely, suppose that E ad P are miimal. Let {ux, u2,..., u} C B be a basis for E ad {vx, v2,..., vm} c C a basis for F. The, {VjU i - Licese or copyright restrictios may apply to redistributio; see

4 2812 SATOSHI KAWAKAMI AND YASUO WATATANI 1,2,...,, j = i,2,..., m} is a basis for Pop by [22, Propositio 1.7.1]. Applyig a characterizatio of the miimal coditioal expectatio i Propositio 2 to E ad P, we have, for x A' C, m m Y^^2(vJ^)x(VjUi)* = «/( KíXK/)»; 1=1 7=1 7=1 1=1 = (Idex F)F(^2 u,xu*) (sice ^ u xu* G P' C by Lemma 1) =i = (Idex P) ^2 u F(x)u* (sice u B ad F : C > B) = (Idex P)(Idex E)E(F(x)) (sice F(x) ea'b). Usig Propositio 2 agai for E o F,we coclude that P o p is miimal. The rest is ow clear, d Combiig Theorem 3 with Remark B(3), we immediately get the followig corollary: Corollary 4 [11], [16], [21]. Let N c L c M be iclusios of factors with fiite idex i Kosaki's sese. The, for ormal coditioal expectatios E : L > N ad F : M -* L, E o F is miimal if ad oly if E ad F are miimal. Moreover, [M : N]o = [M : L]0 [L : N]0. Refereces 1. M. Ballet, Y. Deizeau, ad J. F. Havet, Idice d'ue espérace coditioelle, Compositio Math. 66(1988), A. Coes ad E. Storier, Etropy for automorphisms of II» -vo Neuma algebras, Acta Math. 134(1975), Y. Deizeau ad J. F. Havet, Correspodaces d'idice fii. II: Idice d'ue correspodace, Preprit. 4. F. Goodma, P. de la Harpe, ad V. F. R. Joes, Coxeter Dyki diagrams ad towers of algebras, Spriger-Verlag, Berl, J. F. Havet, Espérace coditioelle miimale, J. Operator Theory 24 (1990), F. Hiai, Miimizig idices of coditioal expectatios oto a subfactor, Publ. Res. Ist. Math. Sei. 24(1988), _, Miimum idex for subfactors ad etropy, J. Operator Theory 24 (1990), M. Izumi, Subalgebras of ifiite C*-algebras with fiite Watatai idices. I: Cutz algebras, Comm. Math. Phys. 155 (1993), V. F. R. Joes, Idex of subfactors, Ivet. Math. 66 (1983), S. Kawakami, Some remarks o idex ad etropy for vo Neuma subalgebras, Proc. Japa Acad. Ser. A Math. Sei. 65 (1989), _, Idicia! derivative for vo Neuma subalgebras, preprit. 12. H. Kosaki, Extesio of Joes ' theory o idex to arbitrary factors, J. Fuct. Aal. 66 ( 1986), H. Kosaki ad R. Logo, A remark o the miimal idex of subfactors, J. Fuct. Aal. 107 (1992), R. Logo, Idex of subfactors ad statistics of quatum fields. I, Comm. Math. Phys. 126 (1989), Licese or copyright restrictios may apply to redistributio; see

5 THE MINIMAL INDEX OF SIMPLE C*-ALGEBRAS _, Idex of subfactors ad statistics of quatum fields. II, Comm. Math. Phys. 130 (1990), _, Miimal idex ad braided subfactors, J. Fuct. Aal. 109 (1992), _, Miimal idex ad uimodular sectors. Quatum ad No-commutative Aalysis, Kluwer, Dordrecht, 1993, pp A. Oceau, Quatized groups, strig algebras, ad Galois theory. Operator Algebras ad Applicatios, Vol. II, Cambridge Uiv. Press, Cambridge, 1988, pp M. Pimser ad S. Popa, Etropy ad idex of subfactors, A. Sei. École Norm. Sup. (4) 19(1986), _, Iteratig the basic costructio, Tras. Amer. Math. Soc. 310 (1988), S. Popa, Classificatio of ameable subfactors of type II, Acta. Math. 172 (1994), Y. Watatai, Idex for C*-subalgebras, Mem. Amer. Math. Soc. No. 424 (1990). 23. S. Yamagami, Modular theory for bimodules, J. Fuct. Aal. 125 (1994), Departmet of Mathematics, Nara Uiversity of Educatio, Nara, 630, Japa address: f61007qsiet.ad.jp Departmet of Mathematics, Kyushu Uiversity, Roppomatu, Fukuoka, 810, Japa address: watatai9math.kyushu-u.ac.jp Licese or copyright restrictios may apply to redistributio; see