A.B. THAHEEM AND NOOR MOHAMMAD. Department of Mathematics Qulad-i-Azam Unlverslt y Islamabad, Pakistan

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1 Iterat. J. Math. & Math. Sci. VOL. 13 NO. (1990) A NOTE ON ONE-PARAMETER GROUPS OF AUTOMORPHISMS A.B. THAHEEM AND NOOR MOHAMMAD Departmet of Mathematics Qulad-i-Azam Ulverslt y Islamabad, Pakista (Received July 7, 1988 ad i revised form September I, 1988) ABSTRACT. Let {t :t R} ad {Bt:t R} be two commutig oe-parameter groups of + Bt + for all B-t *-automorphisms of a yo Neuma algebra M such that a t -t t R. The purpose of this ote is to provide a simple ad short proof of the cetral decompositio result: t t o Mp ad a t B_t projectio p M, without usig the theory of spectral subspaces. o M(l-p) for a cetral KEY WORDS AND PHRASES. Automorphlsm groups, cetral projectios, ivarlat projectios AMS SUBJECT CLASSIFICATION CODES. Primary 46LI0, Secodary 47C15 1. INTRODUCTION I [I], A. Va Daele, L. VaheeswiJck ad oe of the authors proved that if *-automorphlsms of avo Nea algebra M satlsfylg the equatio: = + = t + -t t -t for all t R, the M ca be decomposed to subalgebras Mp ad M(l-p) for o M(l-p), tr. e proof of thls result is very techical ad fairly log. It depeds o veso s theo of spectral subspaces ([2], [3]) ad as such it lacks a proper emphasis o the a cetral projectlo p i M such that t 8t o Mp ad t decompositio itself. However, there are Importat sltuatlos where It Is eough to cosider the comtig automorphlsm groups (see, for Istace [4], [5]). e purpose of thls ote Is to provide a simple proof of thls decompositio result for comtlg automorphlsm groups without usig the theory of spectral subspaces. We use simple algebralc techlques to obtai the proof. Of course, I dolg so we lose the geerality of the result but o the other had we get a simple proof. For more details cocerig the origi of thls operator equatio, Its applicatios ad related dec6osltlo results, we refer to [6], [7], [8], [9], [I0], [4] ad [5].

2 136 A.B. THAHEEM AND N. MOHAMMAD 2. MAIN RESULTS ([i0], The followig is a importat decompositio result for commtlg automorphisms [ii]). PROPOSITION 2.1. Let a,b be commutig *-automorphlsms of a yo Neuma algebra M such that a + a + The there exists a cetral projectio p i M such that a 8 o Mp ad a B o M(l-p). I this paper, however, we use aother versio (Propositio 2.2) of the above result for the sake of clarity. The essetial idea is that whe we cosider the *-automorphism ab o M, the by [12], N(aB- I) + R(aB-1) is o- weakly dese i M where N(aB) ad R(aB) deote, respectlvely, the ull space ad the rage space of the operators uder cosideratio. R(aB) c N(a- B) ad the subalgebra L (say) geerated by R(aB) is a two-sided ideal i M ad there exists a largest cetral projectio P0 i M such that L MP0 MPo c_ N(a- B) ad P0 is a, B- ivarlat (that is a(po) B(po) po ). I other words, a B o Mp 0 (see for istace [10], [II]). Similarly, by cosiderig the orthogoal ideal L i (ote that L i c_ N(aB- I)) we get a largest a,b- Ivarlat cetral projectio q0 i M such that a B o Mqo ad the orthogoallty relatio implies that (I p0)(l qo 0 For more details, we may refer to [10] ad [II]. Thus we may have the followig alterative versio of Propositio 2.1. Propositio 2.2. Let a, 8 be commutig *-automorphlsms of avo Neuma algebra M such that a + a B + 8 Amog the projectios p em (respectively q em) such that a o Mp (respectively a 8 o Mq) there exists a largest cetral projectio PO i M (respectively qo i M) such that P0 ad q0 ivarlat ad (I p0)(l qo O. We ow come to our mai result. THEOREM 2.3. Let {at:t e R} ad {t:t are a, B- e R} be two commutig oe-parameter groups + a 8t + -t for of *-automorphisms of a yo Neuma algebra M such that a t -t all t er. The there exists a cetral projectio p i M such that at(p) t(p) p for all ter ad a t 8 t o Mp ad a t -t PROOF By Propositio 22, let P en, be the largest -ivariat cetral projectio i M such that o M(l p). a 8 o Mp 2-2-

3 ONE-PARAMETER GROUPS OF AUTOMORPHISMS 137 N be the largest a B Similarly, we let q o- 2- i M such that x B o Mq 2-2- The (I -p)(1 -q 0 for ay 6 N. The sequece {p} (+l) 2- (+l) o MP+l -ivariat cetral projectio is decreasig because implies that their squares x ad coiclde o MP+l ad by maximality P+1 ( Similarly the P" sequece {q is decreasig. Put p lim P ad q lira q" The clearly (I-p)(1-q) 0. Sice P is a -ivariat for m, it follows that p is -ivariat for ay 2 -m 2 -m 2- m 2- m m e N. The desity of the set {k 2- m: k Z, m N] i R ad the cotiuity imply that p is t 3 t -ivariat for ay t R. Similarly q is at, Bt (x) (x) for x Mp c_c Agai by the desity of the set k2- m k2- MPm" m {k2- m: k Z, mn}, we coclude that a t B t o Mp for ay t R. Similarly, t B-t completes the proof of the theorem. -ivariat. Also o Mq. As (I q)(l p) 0 the M(I p).c_ Mq ad this ACKNOWLEDGEMENTS. The authors would like to thak Professor Abdus Salam, the Iteratioal Atomic Eergy Agecy ad UNESCO for hospitality at the Iteratioal Cetre for Theoretical Physics, Trieste. REFERENCES I. THAHEEM, A.B., DAELE, A. VAN. ad VANHEESWIJCK, L., A Result o Two Oeparameter Groups of Automorphisms, Math Scad. 51 (1982), 2bI ARVESON, W., O Groups of Automorphisms of Operator Algebras, J. Fuct. Aal._ 15 (1974), DAELE, A. VAN., Arveso s Theory of Spectral Subspaces, Nieuw Arch. Wisk. 27 (1979), HAAGERUP, U. ad SKAU, F., Geometric Aspects of the Tomita-Takesaki Theory II, Math Scad. 48 (1981), HAAGERUP, U. ad OLSEN, H., Tomita-Takesaki Theory for Jorda Algebras, Theory _ll (1984), DAELE, A. VAN., A New Approach to the Tomta-Takesaki Theory of Geeralized Hilbert Algebras, J. Fuct. Aaly J (1974), THAHEEM, A.B., A Bouded Map Associated to a Oe-parameter Group of *- Automorphisms of a yo Neuma Algebra, ow Math. J. _2 (1984), THAHEEM, A.B., O the Operator Equatio a +, +,Iterat. J. Math. & Math. Sci. 9 (1986),

4 138 A.B. THAHEEM AND N. MOHAMMAD 9. CIORANESCU, I. ad ZSIDO,L., Aalytic Geerators for Oe-parameter Groups, Thoku Math. J. (2) 28 (1976) I0. THAHEEM, A. B., Decompositio of a yo Neuma Algebra, Red. Sere. Mat. Uiv. Padova 65(I 981), I-7. II. AWAMI, M. ad THAHEEM, A. B., A Short Proof of a Decompositio Theorem of a yo Neuma Algebra, Proc. Amer. Math. Soc. 92 (1984), THAHEEM, A. B., Decompositio of a yo Neuma Algebra Relative to a *- Automorphtsm, Proc. Ediburgh Math. Soc. (2) 22 (1979), 9-10.

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