A.M. Bikchentaev THE CONTINUITY OF MULTIPLICATION FOR TWOTOPOLOGIES ASSOCIATED WITH A SEMIFINITETRACE ON VON NEUMANN ALGEBRA
(submitted by D. Mushtari)
Key words andphrases. Hilbert space, von Neumann algebra, noncommutative integration,measurable operator, semifinite trace, convergence with respect to measure,compact operator, topological algebra.
Partially supported by Russian Foundation for Basic Research (Grant01-01-00129) and the scientific program ”Universities of Russia – BasicResearch” (Grant UR. 04.01.061) .
ABSTRACT. Let Mbe a semifinite von Neumann algebra in a Hilbert spaceHandτbe a normal faithfulsemifinite trace on M.Let Mprdenote the setof all projections in M,edenote theunit of M, and∥⋅∥denote theC∗-normon M.
The set of all τ-measurableoperators M˜with sum and product defined as the respective closures of the usual sum andproduct, is a *-algebra. The sets
form a base at 0for ametrizable vector topology tτon M˜,called the measure topology. Equipped with this topology,M ˜is a complete topological *-algebra. We will writexi τ →xin case a net{xi}i∈I⊂M˜converges tox∈M˜for the measuretopology on M˜. Bydefinition, a net {xi}i∈I⊂M˜converges τ-locallyto x∈M˜(notation: xi τl →x)if xip τ →xpforall p∈Mpr,τ(p)<∞; and a net{xi}i∈I⊂M˜convergesweak τ-locallyto x∈M˜(notation: xi wτl →x)if pxip τ →pxpforall p∈Mpr,τ(p)<∞.
Theorem 1. Let xi,x∈M˜.
1. If xi τl →x,then xiy τl →xyandyxi τl →yxfor everyfixed y∈M˜.
2. If xi wτl →x,then xiy wτl →xyandyxi wτl →yxfor everyfixed y∈M˜.
Theorem 2. If {xi}i∈I⊂M˜isbounded in measure and if xi τl →x∈M˜,then xiy τ →xyfor allτ-compacty∈M˜.
Theorem 3. Let x,y,xi,yi∈M˜and let a set {xi}i∈Ibebounded in measure. If xi τl →xand yi τl →y,then xiyi τl →xy.
If Mis abelian,then the weak τ-localand τ-localconvergencies on M˜coincides with the familiar convergence locally in measure. Ifτ(e)=∞, thenthe boundedness condition cannot be omitted in Theorem2.
If MisB(H)with standardtrace, then Theorem2 for sequences is a ”Basic lemma”of the theory of projection methods:If yis compactand xn→xstrongly,then xny→xyuniformly, i.e. ∥xny−xy∥→0asn→∞.Theorem 3 means that the mapping
(x,y)↦xy:(B(H)1×B(H)→B(H))
is strong-operator continuous (B(H)1denotes the unit ball of B(H)).
The author is greatly indebted to O.E.Tikhonov for drawing author’s attention to the problemof the τ-localcontinuity of operator functions.
1. Introduction
Let M
be a semifinite von Neumann algebra of operators in a Hilbert space
H and
τ
be a distinguished normal faithful semifinite trace on
M. Let
Mpr denote the lattice
of all projections in M,
e denote the identity,
and M1 denote the
unit ball of M
in the C∗-norm
∥⋅∥ on
M. The closed, densely
defined linear operator x
in H with domain
D(x) is said to be
affiliated with M if
and only if u∗xu=x for all
unitary operators u
in the commutant M′
of M. If
x is affiliated
with M then
x is said to be
τ-measurable if and
only if, for every ɛ>0 there
exists a projection p∈Mpr
for which p(H)⊆D(x) and
τ(e−p)<ɛ. We denote
by M˜ the set of
all τ-measurable
operators. With sum and product defined as the respective closures of the usual sum
and product, M˜
is a *-algebra. The sets
U(ɛ,δ)={x∈M˜:∥xp∥≤ɛandτ(e−p)≤δforsomep∈Mpr},
where ɛ>0,
δ>0, form a base at
0 for a metrizable
vector topology tτ
on M˜,
called the measure topology ([8]; [11, p. 18]). Equipped with this topology,
M˜
is a complete topological *-algebra in which
M is dense. We will
write xi τ →x in case a
net {xi}i∈I⊂M˜ converges to
x∈M˜ for the measure
topology on M˜.
A subset X
of M˜ is boundedin measure, if it is bounded with respect to this topology on the vector space of
M˜, that is in case for
every neighborhood U
of 0 there is
an α>0 such
that αX⊂U
[8, p. 106].
If M
is B(H),
the von Neumann algebra of all bounded linear operators in
H
is equipped with the usual standard trace, then
M˜ coincides
with M
and in this case the measure topology coincides with the
∥⋅∥-topology. If
M is abelian,
then M may be
identified with L∞(Ω,μ)
and τ(f)=∫Ωfdμ where
(Ω,μ) is a localizable measure
space. In this case, M˜
is the space S0(Ω)
consisting of those measurable complex-valued functions on
Ω which
are bounded except on a set of finite measure and the measure topology on
M˜
may be identified simply with the familiar topology of convergence in
measure.
If x is any self-adjoint
operator in H
and if
x=∫ℝλdeλx
is its spectral representation, we will write
χT(x) for the spectral projection
of x corresponding to
the Borel subset T⊂ℝ.
In particular eλx=χ(−∞,λ](x).
If x
is closed, densely defined linear operator affiliated with
M and
∣x∣=x∗x, then the spectral
resolution χ∙(∣x∣) is
contained in M
and x∈M˜ if and only
if there exists λ∈ℝ
such that τ(χ(λ,∞)(∣x∣))<∞.
For p,q∈Mpr we write
p∼q (the Murray - vonNeumann equivalence), if u∗u=p
and uu∗=q for
some u∈M.
A linear set D
in H is said to be
associated with M
if u(D)⊂D for every
unitary operator u
in M′. If
D is a closed linear manifold
then D is associated
with M if and only if
the projection onto D
lies in M [9, p. 403].
For every x∈M˜
the projection onto the closure of the range of
x lies
in M. It
is equal to the left support projection
sl(x)=∧{q∈Mpr:qx=x}
and sl(x)∼sl(x∗).
The two-sided ideal of τ-compact
operators
M˜0={x∈M˜:τ(χ(λ,∞)(∣x∣))<∞forallλ>0}
is closed in measure topology [12]. If
M is
B(H) with standard
trace, then M˜0
is precisely the ideal of compact operators. Let
M0pr=M˜0⋂Mpr={p∈Mpr:τ(p)<∞}.
Definition 1 (cf. [3, p. 114]). A net{xi}i∈I⊂M˜is said toconverge τ-locallyto x∈M˜(notation: xi τl →x)if xip τ →xpforall p∈M0pr.
Definition 2 (cf. [3, p. 114]; [5, p. 746]). A net{xi}i∈I⊂M˜is said to convergeweak τ-locallyto x∈M˜(notation: xi wτl →x)if pxip τ →pxpforall p∈M0pr.
It is clear that
xi τ →x⇒xi τl →x⇒xi wτl →xforxi,x∈M˜.
If M is
B(H) with standard trace,
then τ-local (respectively,
weak τ-local)
convergence coincides with strong-operator (respectively, weak-operator) convergence.
If τ(e)<∞,
then M˜
consists of all densely defined closed linear operators affiliated with
M and weak
τ-local
convergence is precisely the convergence in measure topology on
M˜.
Moreover, the measure topology is a minimal one in the class of all topologies
which are Hausdorff, metrizable, and compatible with the ring structure of
M˜ [1,
Theorem 2].
2. Main Results
Further we assume that τ(e)=∞.
Theorem 1. Let xi,x∈M˜.
1. If xi τl →x,then xiy τl →xyandyxi τl →yxfor everyfixed y∈M˜.
2. If xi wτl →x,then xiy wτl →xyandyxi wτl →yxfor everyfixed y∈M˜.
Proof. Let xi,x,y∈M˜
and let p∈M0pr.
Since sl(yp)∼sl(py∗)≤p,
one has sl(yp)∈M0pr.
1. Suppose that xi τl →x.
One has
yxi τl →yxandxiyp=xisl(yp)yp τ →xsl(yp)yp=xyp,
since the multiplication operations z↦yz(M˜→M˜)
and z↦zyp(M˜→M˜)
are continuous in the measure topology.
2. One has r=p∨q∈M0pr
for p,q∈M0pr, since
p∨q−p∼q−p∧q [8, p. 105]. By
[3, p. 114] xi wτl →x if
and only if pxiq τ →pxq
for all p,q∈M0pr.
Indeed, from rxir τ →rxr
it follows that
pxiq=p⋅rxir⋅q τ →p⋅rxr⋅q=pxq.
Therefore,
pxiyp=pxisl(yp)yp τ →pxsl(yp)yp=pxyp.
Now the convergence yxi wτl →yx
follows from the fact that the mapping
z↦z∗(M˜→M˜) is weak
τ-local
continuous and by taking adjoints.
Theorem 2. If {xi}i∈I⊂M˜isbounded in measure and if xi τl →x∈M˜,then xiy τ →xyforall y∈M˜0.
Proof. Step 1. Without loss of generality we may assume that
y∈M˜0 is self-adjoint and
non-negative. Indeed, let y∈M˜0
and y∗=u∣y∗∣ be the polar
decomposition of y∗.
Then y=∣y∗∣u∗ and
from xi∣y∗∣τ →x∣y∗∣ it follows
that xiy τ →xy, since the
multiplication operation z↦zu∗(M˜→M˜)
is continuous in the measure topology.
Step 2. Fix non-negative y∈M˜0
and ɛ,δ>0. A
subset X
of M˜
is bounded in measure if and only if for every
d>0 there exists
a constant c<∞
such that X⊂U(c,d) [8,
p. 106]. Let n∈ℕ
and
by [8, p. 107], [11, p. 18]. The assertion of Theorem 2 follows from
(1),
(5) and
(6),
since
xiy−xy∈U(3ɛ,3δ)foralli∈I,i≥i0.
Theorem 3. Let x,y,xi,yi∈M˜and let a set {xi}i∈Ibebounded in measure. If xi τl →xand yi τl →y,then xiyi τl →xy.
Proof. For every p∈M0pr
one has
xiyip−xyp=xi(yip−yp)+(xi−x)yp,i∈I.(8)
Fix ɛ,δ>0.
By assumption of the theorem, there exists a constant
c>0 such
that
{xi}i∈I⊂U(c,δ).(9)
Since yip−yp τ →0,
there exists i1∈I
such that
yip−yp∈U(2ɛc−1,δ)foralli∈I,i≥i1.(10)
Now by (9),
(10) and
(4) one
has
xi(yip−yp)∈U(2ɛ,2δ)foralli∈I,i≥i1.(11)
Since xi−x τl →0 and
yp∈M˜0 , it follows by Theorem
2 that there exists i2∈I
such that
(xi−x)yp∈U(ɛ,δ)foralli∈I,i≥i2.(12)
There exists i0∈I
such that i0≥i1
and i0≥i2.
Now by (8),
(11),
(12) and
(7) one
has
xiyip−xyp∈U(3ɛ,3δ)foralli∈I,i≥i0.
This proves the theorem.
Example 1. If M is abelian,
then the weak τ-local
and τ-local
convergencies on M˜
coincides with the familiar convergence locally in measure (i.e., in other words,
convergence in measure on every set of finite measure). The boundedness condition
for {xi}i∈I
cannot be omitted in Theorem 2. Indeed, let
Ω=(0,∞) equipped with the
Lebesgue measure μ.
Define the functions
y(t)= min{1,t−1};xn(t)=tχ[n,2n](t)(t∈(0,∞),n∈ℕ).
Then
i) xn τl →0
as n→∞;
ii) {xn}n=1∞
is not bounded in measure;
iii) y∈M˜0⋂M1;
iv) since (xny)(t)=χ[n,2n](t)
for every t∈(0,∞),n∈ℕ,
xny does
not converge in measure topology.
Example 2. If M
is B(H)
with standard trace, then Theorem 2 for sequences is a ”Basic lemma”of
the projection methods [2, pp. 18–19] (the boundedness condition for
{xn}n=1∞
follows from the principle of uniform boundedness):
If yiscompact and xn→xstrongly, then xny→xyuniformly, i.e. ∥xny−xy∥→0asn→∞.
Theorem 3 means that the mapping
(x,y)↦xy:(B(H)1×B(H)→B(H))
is strong-operator continuous [7, pp. 115–117].
Remark. The second part of Theorem 1 was already used in [6] and
[10].
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