PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY Volume 102, Number 4, April 1988
Lp-CONTINUITY OF POSITIVE SEMIGROUPS ON FINITE VON NEUMANN ALGEBRAS
SEIJI WATANABE (Communicated by John B. Conway)
ABSTRACT. Let M be a <7-finite, finite von Neumann algebra with a faithful normal tracial state t. Let a be a one-parameter semigroup of normal positive contractions of M. Then it is shown that a is continuous with respect to the Lp-norm (1 < p < oo) induced by t if and only if it is cr-weakly continuous.
1. Introduction. Various useful topologies are introduced on von Neumann algebras, including the (operator) norm topology. Hence, various continuity proper- ties can also be defined for one-parameter semigroups of linear operators. However, only cr-weakly continuous semigroups or groups have been discussed in most pa- pers, except uniformly continuous ones. Indeed, R. R. Kallman [5, 6] showed that every strongly continuous (namely, continuous in the pointwise norm convergence topology) one-parameter group of *-automorphisms of von Neumann algebras is already uniformly continuous, and G. A. Elliott [2] showed the same result for AW*-algebras. Furthermore, the uniform continuity of strongly continuous posi- tive semigroups was shown by A. Kishimoto and D. W. Robinson [7] for abelian von Neumann algebras and by U. Groh [4] for properly infinite von Neumann algebras.
Nevertheless, the problem for arbitrary von Neumann algebras remains open.
In this paper, we examine Lp-continuity of positive semigroups on finite von Neumann algebras, using a technique of [3, 8].
2. Result. Let M be a tr-finite, finite von Neumann algebra with a fixed faithful normal tracial state r. Let || ||p (1 < p < oo) be the Lp-norm on M defined by ||z||p = r(|x|p)1/,p (x E M) and || ||oo be the (operator) norm on M. Let a = {at}t>o be a one-parameter semigroup of normal positive contractions on M with ao — I (the identity operator). Then a is said to be Lp-norm continuous if limt|o IJCKt(a:) — x||p = 0 for every x E M (1 < p < oo).
We shall consider Lp-continuity of a in the following. If a is Lp-norm continuous for some p (1 < p < oo), then a is cr-weakly continuous. Indeed, since r is faithful and every at is contraction with respect to || ||oo, |r(at(a:) — %)\ ^ llat(x) — x\\p (x E M) implies that p(at(x)) -* p(x) (t —> +0) for every x E M and p E M, (the predual space of M). The converse statement does not hold for p = oo in general as mentioned above. But it does hold for p (1 < p < oo). Namely, we have the following theorem. This is interesting in connection with the problem mentioned in the Introduction and the fact that limp_00 ||x||p = Hx^ for x E M.
Received by the editors December 19, 1986.
1980 Mathematics Subject Classification (1985 Revision). Primary 46L10, 46L50, 47D05.
Key words and phrases. Positive semigroup, trace, finite von Neumann algebra.
©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page 840
Lp-CONTINUITY OF POSITIVE SEMIGROUPS 841 THEOREM. Let M,r, and a be as above. Then a is Lp-norm continuous for every p (1 < p < oo) if and only if a is a-weakly continuous.
Before going to the proof, we should note that r is not necessarily assumed to be a-invariant.
PROOF OF THEOREM. The if part is already seen. Conversely, suppose that a is cr-weakly continuous. Let ß = {ßt)t>o be the strongly continuous contraction semigroup on L1(M,t) corresponding to the preadjoint semigroup of a on TW«, where Ll(M,r) is the noncommutative Z^-space realized as the space of closed operators [1, 9]. We define two positive elements tç> e M* and h E L1(M,t) by
/•OO
tq = I e~lr o at dt and
L
e-*ßt(l)dt.
Then To is faithful and h is injective because ao — I and rn(-) = r(h-). Now, an inner product ( , ) on M is defined by (x, y) = To(y*x + xy*) (x,y E M), and let H be the completion of M in the norm ||x¡2 = (x, x)1/2 (x € M). Next, we put 7t(x) = e~lat(x) (x E M, t > 0). Then 7 = {7i}t>o is a a-weakly continuous semigrup of normal positive contractions of M, and we have, for x E M,
/■OO
To(-jt(x*x + xx*)) = / e~sT(as(e~tat(x*x + xx*)))ds
Jo
/oo e~sT(as(x*x + xx*)) ds
/■OO
< / e~sT(as(x*x + xx*))ds
Jo
= To(x*X + XX*).
Since each qt is a positive || H^-contraction on M, we have, for x E M,
ht\f2=ro(lt(xrit(x)+lt(x)lt(xr)
< To(7¿(x*x + xx*))
< r0(x*x + xx*) = ¡xi2,.
That is, each 7t is contraction on M with respect to the norm ||| ¡2. Hence, each 7t extends to a contraction, say 7t, on H. Thus, we have a contraction semigroup on H which is weakly continuous because 7 is cr-weakly continuous on M. Hence, 7 = {7t}t>o is strongly continuous on H, that is,
Um|7*(a;)-x|2 = 0 (xeH).
By a simple calculation, we have lim(|0 ||£*t(z) — x\2 = 0 (x E M). Next, let h = j0°° Xde(X) be the spectral resolution of h. Then r(e(X)) —► 7"(e(0)) = 0 by the injectivity of h. Therefore, for each positive number e > 0, there exists a positive number A > 0 such that r(e(X)) < e. Then we have A(l - e(A)) < h. Thus, we
842 SEIJI WATANABE
have, for x E M,
\\at(x) -x\\l = r(|ai(x) - x|2)
= r((l - e(X))\at(x) - x|2) + r(e(A)|aÉ(x) - x|2)
= T(\at(x)-x\(l-e(X))\at{x)-x\) + r(e(A)|ai(2:)-x|2e(A))
< (l/A)7-(|at(x) - x\h\at(x) - x|) + (2||x||00)2r(e(A))
< (l/X)T(h\at(x) - x\2) + (2\\x\U2e
= (l/A)r0(|ai(x)-x|2) +411x11^6
<(l/A)|at(x)-x||2+411x11^.
Since ¡«¿(x) — x||2 —► 0 (t J. 0) and e is arbitrary, we have limtjo ||at(x) — x||2 = 0 (x E M). Furthermore, inequalities ||a||p < ||a||oo~1)/p||a||2/P (a E M, oo > p > 1) imply our assertion. The proof is completed.
REFERENCES
1. J. Dixmier, Formes linéaires sur un anneau d'operateurs, Bull. Soc. Math. France 81 (1953),
9 39.
2 G. A. Elliott, Convergence of automorphisms in certain C -algebras, J. Funct. Anal. 11 (1972),
204-206.
3. R. Emilion, Semi-groups in Loo and local ergodic theorem, Cañad. Math. Bull. 29 (1986), 146-153.
4. U. Groh, Positive semi-groups on C*~- and W* -algebras, Lecture Notes in Math., vol. 1184, Springer-Verlag, Berlin and New York, 1986, pp. 369-425.
5. R. R. Kallman, Unitary groups and automorphisms of operator algebras, Amer. J. Math. 91
(1969), 785-806.
6. _, One-parameter groups of " -automorphisms of IIi von Neumann algebras, Proc Amer.
Math. Soc, 24 (1970), 336-340.
7. A. Kishimoto and D. W. Robinson, Subordinate semi-groups and order properties, J. Austral.
Math. Soc. Ser. A 31 (1981), 59-76.
8. _, On unbounded derivations commuting with a compact group of '-automorphisms, Publ.
Res. Inst. Math. Sei. 18 (1982), 1121-1136.
9. I. Segal, A noncommutative extension of abstract integration, Ann. of Math. (2) 57 (1953),
401-457.
Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-21, Japan
