Bull. Kyushu Inst. Tech.
(Math. Natur. Sci.)'No, 33, 1986, pp. 1-3
A NOTE ON CONTRACTION SEMIGROUPS IN FINITE VON NEUMANN ALGEBRAS
By
Shigeru IToH
(Received November 8, 1985)
In [4], Foias'and Kovacs gave a necessary and sufficient condition for a contraction in a finite von Neumann algebra to be a unitary operator. Then, in [8], Kovacs obtained a similar condition for a one-parameter semigroup of contractions belonging to a finite von Neumann algebra to be a (positive) part of a one-parameter group of unitary operators.
In this note we give an analogous result for a general semigroup of contractions with some appropriate conditions.
Now let S be a semigroup. If for any s, teS, SsnSt#O, then S is called right reversible, where Ss={us: ueS}. Right amenable semigroups and, in particular, abelian semigroups are right reversible. For a right reversible semigroup S, define "/År " by t!Års if and only if t=s or teSs (s, tES). Then S becomes a directed set.
Let H be a Hilbert space and B(H) be the set of all bounded linear operators on H.
Let Åë: S.B(H) be a homomorphism. IfS is right reversible, then {Åë,},.s is a net in B(H) by considering S as a directed set defined as above. We say that Åë belongs to the class Ci. (denoted by ÅëeCi.) if s-lim Åë,(4)=O for 4=O (4EH) only, where s-lim is the limit
tin the strong topology.
We have the following result.
THEoREM. Let S be a right reversible semigroup, M a finite von Neumann algebra on H, and Åë: S.M a homomorphism such that for any teS, di, is a contraction. Then ÅëECi. if and only if for any tES, Åë, is a unitary operator.
PRooF. The method of the proof is a slight modification of those in [4], [8]. Let ÅëE Ci. . For each te S, let ut,= ÅëfÅë,, then {gef,},.s is a net in M. If t=us, where s, t, u E S,
then for any 4eH,
O `Å~ (9e't(4), 4)= IIÅët(4)ll2 = llÅëuÅës(`li)Il2 ÅqÅ~ ll Åë,(4) ll 2 = : ( V.((li), C) •
Thus, the net {Y,},.s is decreasing, and converges to some aeM in the strong operator topology. It follows that a is positive and one to one. In fact, if a(4)=O, then
lim ll Åë,(4) 11 2 - lim ( Y,(4), 4) - (a(4), 4) - O.
We have 4=O because ÅëeCi. . Since S is right reversible, for each sES, {!I'.}..s, is a subnet of {!l/,},.s, hence converges to a in the strong operator topology too. For every
,te S, Åë," Y',Åë,= Y,., thus, in the limit we have di,"aÅë,=a. The rest of the proof is as in [8].
For completeness we continue the proof. Let ip be any finite normal trace on M. Then
ip(a) = ip(Åë,*aÅë,) = ip(a i!2Åë.Åë,*ai/2) ,
hence ip(ai/2(I--Åë,di,")a'!2)=O, where I is the identity operator on H. As M is finite, we obtain a`12(I-di,Åë.")a'/2=O (cf. [2, Definition I.6.7.5], [10, Theorem V.2.4]). Since a is positive and one to one, ai12 is also one to one and the range of ai12 is dense in H, so we have I-- Åë,di,"=O. The finiteness ofM implies that Åë,* i's unitary, and so is di, (cf.
[2, Proposition III.1.1.3], [10, Definitions V.1,2 and 1.15]).
The converse is obvious.
Now let S be an abelian semigroup with the unit O (whose semigroup operation is denoted by +). In the direct product abelian semigroup S Å~ S, we define the equivalence relation fs as follows: For (s, t), (u, v)ESÅ~ S, (s, t)N(u, v) if and only if there exists an
element weS such that s+v+w= t+u+w. Let K(S) be the set of equivalence classes of SxS, then K(S) naturally becomes an abelian group. K(S) is called the 6rothendieck group of S. For (s, t)ESÅ~S, denote by [s, t] the equivalence class of SÅ~S containing (s, t). Define 7: S.K(S) by y(s)=[s, O] (sES). Then 7 is a homomorphism.
The following is an extension of results in [4], [8]. (Related results are treated in [3], but it seems to me that the statement of Definition 3.3 is curious, hence so are proofs of Theorems 3.4 and 3.6 in [3].)
CoRoLLARy. Let S be an abelian semigroup with O, M a finite von Neumann algebra on H, and Åë: S.M a homomorphism such that for every te S, di, is a contraction. Then dieCi. if and only if there exists a homomorphism V: K(S).M such that for every
g e K(S), Y', is unitary and for each t e S, Y',(,) == di,.
PRooF. Let ÅëeCi., Then, by Theorem, Åë, is unitary for all tES. Define !U:
K(S)--hM by !U,=:Åë,Åëf forg=[s, t]eK(S). Then !P is well-defined. In fact, if (s, t)sy (u, v), then there exists weS such that s+v+w=t+u+w. It follows that
ÅësÅët* --""- ÅësÅëvÅëif Åëwdi:ÅëuÅë:Åëf
=dis+v+w(Åët+u+w)*Åëudi:
= ÅëuÅë:'
A Note on Contraction Semigroups in Finite Von Neumann Algebras 3
Moreover, for any t E S, Y',(t) = ÅëtÅëo* = Åët•
The converse direction is evident.
References
[1] A.H.Clifford and G.B.Preston, The Algebraic Theory of Semigroups, 2 Volumes, Amer.
Math. Soc., 1981.
[2] J. Dixmier, Von Neumann Algebras, North-Holland, 1981.
[ 3 ] D. E. Evans and J. T. Lewis, Dilations of Irreversible Evolutions in AIgebraic 'Quantum Theory, Dublin Institute for Advanced Studies, 1977.
[4] C. Foias and I. Kovacs, Une caracterisation nouvelle des algebres de von Neumann finies, Acta Sci. Math. 23 (1962), 27`F278.
[ 5 ] F. P. Greenleaf, Invariant Means on Topological Groups and Their Applications, Van Nostrand, 1969.
[6] S. Itoh, A note on dilations in modules over C"-algebras, J. London Math. Soc. (2) 22 (1980), 117-126.
[ 7 ] J. L. Kelley, General Topology, Van Nostrand, 1955.
[8] I. Kovacs, Dilation theory and one-parameter semigroups of contractions, Acta Sci. Math.
45 (1983), 279-280. -[9] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, 1970.
[10] M. Takesaki, Theory of Operator Algebras l, Springer, 1981.
[11] D.M.Topping, LecturesonvonNeumannAlgebras, VanNostrand,1971.
Department of Mathematics, Kyushu Institute of Technology
