Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 14, 2004, 17–24

© A.M. Bikchentaev

ABSTRACT. Let M be a semifinite von Neumann algebra in a Hilbert space H and τ be a normal faithful semifinite trace on M. Let Mpr denote the set of all projections in M, e denote the unit of M, and ∥ ⋅ ∥ denote the C∗ -norm on M.

The set of all τ-measurable operators M˜ with sum and product defined as the respective closures of the usual sum and product, is a *-algebra. The sets

form a base at 0 for a metrizable vector topology tτ on M˜, called the measure topology. Equipped with this topology, M ˜ is a complete topological *-algebra. We will write xi τ → x in case a net {xi }i∈I ⊂M˜ converges to x ∈M˜ for the measure topology on M˜. By definition, a net {xi}i∈I ⊂M˜ converges τ-locally to x ∈M˜ (notation: xi τl → x) if xip τ → xp for all p ∈Mpr, τ(p) < ∞; and a net {xi }i∈I ⊂M˜ converges weak τ-locally to x ∈M˜ (notation: xi wτl → x) if pxip τ → pxp for all p ∈Mpr, τ(p) < ∞.

Theorem 1. Let xi,x ∈M˜.

1. If xi τl → x, then xiy τl → xy and yxi τl → yx for every fixed y ∈M˜.

2. If xi wτl → x, then xiy wτl → xy and yxi wτl → yx for every fixed y ∈M˜.

Theorem 2. If {xi}i∈I ⊂M˜ is bounded in measure and if xi τl → x ∈M˜, then xiy τ → xy for all τ-compact y ∈M˜.

Theorem 3. Let x,y,xi,yi ∈M˜ and let a set {xi}i∈I be bounded in measure. If xi τl → x and yi τl → y, then xiyi τl → xy.

If M is abelian, then the weak τ-local and τ-local convergencies on M˜ coincides with the familiar convergence locally in measure. If τ(e) = ∞, then the boundedness condition cannot be omitted in Theorem 2.

If M is B(H) with standard trace, then Theorem 2 for sequences is a ”Basic lemma”of the theory of projection methods: If y is compact and xn → x strongly, then xny → xy uniformly, i.e. ∥xny − xy∥→ 0 asn →∞. Theorem 3 means that the mapping

is strong-operator continuous (B(H)1 denotes the unit ball of B(H)).

The author is greatly indebted to O.E.Tikhonov for drawing author’s attention to the problem of the τ-local continuity 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

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 bounded in 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

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 - von Neumann 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

and sl(x) ∼ sl(x∗).

The two-sided ideal of τ-compact operators

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

Definition 1 (cf. [3, p. 114]). A net {xi }i∈I ⊂M˜ is said to converge τ-locally to x ∈M˜ (notation: xi τl → x) if xip τ → xp for all p ∈M0pr.

Definition 2 (cf. [3, p. 114]; [5, p. 746]). A net {xi }i∈I ⊂M˜ is said to converge weak τ-locally to x ∈M˜ (notation: xi wτl → x) if pxip τ → pxp for all p ∈M0pr.

It is clear that

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 → xy and yxi τl → yx for every fixed y ∈M˜.

2. If xi wτl → x, then xiy wτl → xy and yxi wτl → yx for every fixed 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

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

Therefore,

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˜ is bounded in measure and if xi τl → x ∈M˜, then xiy τ → xy for all 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

Then y = y1,n + y2,n + y3,n and for zi = xi − x one has

The set {zi}i∈I is bounded in measure. There exists a constant c > 0 such that

Let

Since ∥y1,n∥ < n−1, one has y1,n τ → 0 as n →∞. Since

one has y3,n τ → 0 as n →∞. Therefore y1,n + y3,n τ → 0 as n →∞. Then there exists m ∈ ℕ such that

Recall that

by [8, p. 107], [11, p. 18]. Now by (2) and (3) one has

Step 3. Let m be as above, λk > 0 and pk ∈M0pr (k = 1,…,j), pk pl = 0 for k≠l, such that

(one can choose pk as spectral projections of y). There exists z ∈M1 such that

[4, Chap. 1, Sect. 1, Lemma 2]. Since zi τl → 0, one has zi pk τ → 0 for all k = 1,…,j. Now

because the multiplication operation t↦tz (M˜→M˜) is continuous in the measure topology. Therefore, there exists i0 ∈ I such that

Step 4. Recall that

by [8, p. 107], [11, p. 18]. The assertion of Theorem 2 follows from (1), (5) and (6), since

Theorem 3. Let x,y,xi,yi ∈M˜ and let a set {xi}i∈I be bounded in measure. If xi τl → x and yi τl → y, then xiyi τl → xy.

Proof. For every p ∈M0pr one has

Fix ɛ,δ > 0. By assumption of the theorem, there exists a constant c > 0 such that

Since yip − yp τ → 0, there exists i1 ∈ I such that

Now by (9), (10) and (4) one has

Since xi − x τl → 0 and yp ∈M˜0 , it follows by Theorem 2 that there exists i2 ∈ I such that

There exists i0 ∈ I such that i0 ≥ i1 and i0 ≥ i2. Now by (8), (11), (12) and (7) one has

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

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, xn y 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 y is compact and xn → x strongly, then xny → xy uniformly, i.e. ∥xny − xy∥→ 0 asn →∞.

Theorem 3 means that the mapping

is strong-operator continuous [7, pp. 115–117].

Remark. The second part of Theorem 1 was already used in [6] and [10].

References

RESEARCH INSTITUTE OF MATHEMATICS AND MECHANICS, KAZAN STATE UNIVERSITY, UNIVERSITETSKAYA STR. 17, KAZAN:420008, RUSSIA

E-mail address: Airat.Bikchentaev@ksu.ru

Received January 8, 2004