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APPEARED IN BULLETIN OF THE

arXiv:math/9204228v1 [math.OA] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 288-293THE MACKEY-GLEASON PROBLEML. J. Bunce and J. D. Maitland WrightAbstract.

Let A be a von Neumann algebra with no direct summand of Type I2,and let P(A) be its lattice of projections. Let X be a Banach space.

Let m: P(A) →X be a bounded function such that m(p + q) = m(p) + m(q) whenever p and q areorthogonal projections. The main theorem states that m has a unique extension to abounded linear operator from A to X.

In particular, each bounded complex-valuedfinitely additive quantum measure on P(A) has a unique extension to a boundedlinear functional on A.Physical backgroundIn von Neumann’s approach to the mathematical foundations of quantum me-chanics, the bounded observables of a physical system are identified with a reallinear space, L, of bounded selfadjoint operators on a Hilbert space H. It is reason-able to assume that L is closed in the weak operator topology and that wheneverx ∈L then x2 ∈L. (Thus L is a Jordan algebra and contains spectral projections.

)Then the projections in L form a complete orthomodular lattice, P, otherwiseknown as the lattice of “questions” or the quantum logic of the physical system. Aquantum measure is a map µ: P →R such that whenever p and q are orthogonalprojections µ(p + q) = µ(p) + µ(q).In Mackey’s formulation of quantum mechanics [11] his Axiom VII makes theassumption that L = L(H)sa.

Mackey states, that in contrast to his other axioms,Axiom VII has no physical justification; it is adopted for mathematical convenience.One of the technical advantages of this axiom was that, by Gleason’s Theorem, acompletely additive positive quantum measure on the projections of L(H) is therestriction of a bounded linear functional (provided H is not two-dimensional). Inorder to weaken Axiom VII it was desirable to strengthen Gleason’s Theorem.IntroductionLet P(A) be the lattice of projections in a von Neumann algebra A, let X be aBanach space, and let µ: P(A) →X be a function such that(a) µ(e + f ) = µ(e) + µ(f ) whenever ef = 0,(b) sup{∥µ(e)∥: e ∈P(A)} < ∞.1991 Mathematics Subject Classification.

Primary 46L50.Received by the editors May 21, 1991c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2L. J. BUNCE AND J. D. MAITLAND WRIGHTThen µ is said to be a finitely additive, X-valued measure on P(A).Clearly each bounded linear operator from A to X restricts to a finitely ad-ditive X-valued measure.

When A is the algebra of two-by-two matrices and Xis one-dimensional, there exist examples of measures that fail to extend to linearfunctionals.Our main result isTheorem A. Let A be a von Neumann algebra with no direct summand of TypeI2.

Then, for each Banach space X, each X-valued measure on P(A) has a uniqueextension to a bounded linear operator from A to X.This immediately specializes to giveTheorem B. Let A be a von Neumann algebra with no direct summand of Type I2.Then each complex-valued finitely additive measure on P(A) extends to a boundedlinear functional on A.In fact, we shall see that Theorem A follows easily from Theorem B.

We note,however, that to deduce Theorem A it is essential to have Theorem B for all real-valued finitely additive measures. It does not suffice to know this result for positivemeasures or for countably additive measures.

The lack of positivity causes consid-erable difficulty in establishing this theorem.Theorem B answers a natural question first posed by G. W. Mackey some thirtyyears ago. When µ is positive, that is, when µ(e) ≥0 for each projection e. TheoremB was established by Christensen [7] for properly infinite algebras and algebras ofType In and by Yeadon [15, 16] for algebras of finite type.

The first major progresshad been made by Gleason [9], who, by using an ingenious geometric argument,settled the question for positive completely additive measures on the projections ofL(H). (See [8] for an elementary proof of this deep result.) Aarnes [1] and Gun-son [10] made important contributions, especially concerning continuity properties.Paszkiewicz [13], working independently of Christensen and Yeadon, establishedTheorem B for σ-finite factors, however, to extend his results to nonfactorial vonNeumann algebras, he requires µ to be positive and countably additive.

A lucidand meticulous exposition of Theorem B for positive measures is given by Maeda[12].1. Vector measuresThe following short argument shows that Theorem A is a consequence of Theo-rem B.Lemma 1.1.

Let A be a von Neumann algebra such that each finitely additive(complex) measure on P(A) has an extension to a bounded linear functional. Let Xbe a Banach space and let m: P(A) →X be a finitely additively X-valued measure.Then m has a unique extension to a bounded linear operator from A to X.Proof.

Elementary estimates show that when β ∈A∗then∥β∥≤4 sup{|β(p)|: p ∈P(A)} .Let K be a constant such that ∥m(p)∥≤K for each p ∈P(A).For any φ ∈X∗, p →φm(p) is a complex-valued finitely additive measure onP(A). By hypothesis there exists β ∈A∗such that β(p) = φm(p) for each p ∈P(A).

THE MACKEY-GLEASON PROBLEM3Let x = Pn1 λjpj be a finite linear combination of projections p1, p2, . .

. , pn.

Thenφ nX1λjm(pj)!=nX1λjφm(pj) = β nX1λjpj!= β(x) .Soφ nX1λjm(pj)! ≤4∥x∥∥φ∥sup{∥m(p)∥: p ∈P(A)}≤4∥x∥∥φ∥K .It follows from the Hahn-Banach Theorem thatnX1λjm(pj) ≤4K∥x∥.In particular, Pn1 λjpj = 0 implies Pn1 λjm(pj) = 0.

So m has a unique extensionto a linear operator T : Span P(A) →X.T nX1λjpj! =nX1λjm(pj) ≤4KnX1λjpj .So T is bounded and hence has a unique extension to a bounded linear operatorfrom A to X.2.

Scalar measuresIn all that follows A is a von Neumann algebra with no direct summand of TypeI2. Let P(A) be the lattice of projections in A.Since each complex-valued measure on P(A) is of the form µ+iν, where µ and νare real-valued measures, it suffices to prove Theorem B for real-valued measures.From now onward, µ is a finitely additive real-valued measure on P(A).

Thatis, µ: P(A) →R is a function such that(a) µ(p + q) = µ(p) + µ(q) whenever p and q are orthogonal projections;(b) sup{|µ(p)|: p ∈P(A)} < +∞.We define the variation of µ, V , byV (p) = sup{|µ(e)|: e ≤p};we also defineα(p) = sup{µ(e): e ≤p} .Straightforward arguments show that µ has a unique extension to a functionµ: A →C, where µ is linear and bounded on each abelian ∗-subalgebra of A andwhere µ(x + iy) = µ(x) + iµ(y) whenever x and y are selfadjoint. Moreover, it canbe shown [4] that(1) sup{|µ(x)|: x = x∗and ∥x∥≤1} = 2α(1) −µ(1),(2) sup{µ(x): 0 ≤x ≤1} = α(1).

4L. J. BUNCE AND J. D. MAITLAND WRIGHTIn order to see (2), let 0 ≤x ≤1.Then, for suitable spectral projections,x = P∞1 2−nen.

Hence, for some n, µ(en) ≥µ(x).The lack of positivity of µ greatly increases the difficulties of establishing thelinearity of µ when A is properly infinite or when A is of Type II1. However, whenA is of Type In (n ̸= 2) linearity can be established by a fairly straightforwardextension of the arguments for positive measures.The first step is to notice that when µ is a measure on the projections of Mn(C),the algebra of n × n matrices over C, then when T is the canonical (unnormalized)trace on Mn(C) we have µ(e) ≤α(1) = α(1)T (e) for each minimal projection e. Soα(1)T −µ is a positive measure on P(Mn(C)).

Hence, by Gleason’s Theorem forfinite-dimensional Hilbert spaces, α(1)T −µ is linear (provided n ̸= 2). We nowrevert to the general situation and conclude that µ is linear on each subalgebraof A that can be embedded in a subalgebra of A that is isomorphic to a Type Infactor (n ≥3).

By elementary algebraic arguments, it can be shown [8] that if B isa subalgebra of A and B ≈M2(C) ⊂D, then either B ⊂C ⊂A, where C ≈M4(C)or B ⊂C ⊕D, where C ≃M4(C) and D ⊂E ⊂A with E ≈M3(C). Hence µ islinear on B.By “patching” together Type I2 factorial subalgebras of A it can be shown [2],building on techniques of Christensen [7], that µ is uniformly continuous on P(A)and µ is linear on each Type In subalgebra of A.In particular, µ is linear onW(1, p, q), where W(1, p, q) is the W ∗-subalgebra of A generated by the identityand an arbitrary pair of projections p and q.3.

Approximate linearityThe restriction of µ to the centre of A is linear. So, by the Hahn-Banach Theo-rem, there exists σ ∈A∗such that µ and σ coincide on the centre.

By replacing µby µ−σ if necessary, we can assume that µ vanishes on the centre of A. By dividingby a suitable constant we can also assume that sup{|µ(x)|: x = λ∗, ∥λ∥≤1} = 1.It can then be shown that α(1) = 12.Lack of positivity leads to a number of difficulties establishingLemma 3.1.

Let A be properly infinite. Let 0 < δ < 12.

There exists a projectione in A with 1 ∼e ∼1 −e and such that 12 −δ2 < µ(e). Then, for each p ∈P(A).|µp −µ(epe) −µ(1 −e)p(1 −e)| < 5δ .The next lemma depends on the fact that µ is linear on each W ∗-subalgebragenerated by a pair of projections.Lemma 3.2.

Let 0 < ε < 1 and let m be such that P∞m+1 2−n < ε. Let e be aprojection such that(1) |µ(p) −µ(epe) −µ((1 −e)p(1 −e))| < ε/m for each p ∈P(A) ;(2) |µ(a + b) −µ(a) −µ(b)| < ε/m whenever a ≥0, b ≥0, and a + b ≤e ;(3) |µ(c + d) −µ(c) −µ(d)| < ε/m whenever c ≥0, d ≥0, and c + d ≤1 −e.Then, whenever x ≥0, y ≥0, and x + y ≤1, |µ(x + y) −µ(x) −µ(y)| < 20ε.We shall now sketch a proof of Theorem B for A a properly infinite von Neumannalgebra.Let a and b be fixed, positive elements of A with a + b ≤1.

Choose ε with0 < ε < 1. Let m be such that P∞m+1 2−n < ε.

We put δ = ε/42m. Let e be aprojection that satisfies the conditions of Lemma 3.1.

THE MACKEY-GLEASON PROBLEM5Let x ≥0, y ≥0 with x + y ≤e.We find a projection f ≤1 −e such that f ∼1 and V (f ) < δ2. Put h = e + f.We then find three orthogonal projections, majorized by f and each equivalent toe.

By applying the 4 × 4 matrix construction due to Christensen [7] we can findorthogonal projections p and q, majorized by h = e+f, with x = 2epe and y = 2eqe.Since x is in the ∗-algebra generated by e and p,|2µ(p) −µ(x)| = |µ(2p −x)| = 2|µ(p −epe)| .Since e, f, and p are in the properly infinite W ∗-algebra hAh and e + f = h, wesee that e is in the W ∗-subalgebra of hAh generated by f and p. So2|µ(p −epe)| ≤2|µ(fpf )| + 2|µ(fpe + epf )| .By considering µ restricted to fAf we can show that|µ(fpf )| ≤2V (f ) < 2δ2 .Also,fpe + epf = (1 −e)pe + ep(1 −e) .So, applying Lemma 3.1, |µ(fpe + epf )| < 5δ. So|2µ(p) −µ(x)| < 2δ2 + 10δ < 14δ .Similarly,|2µ(q) −µ(y)| < 14δand|2µ(p + q) −µ(x + y)| < 14δ .So|µ(x + y) −µ(x) −µ(y)| < 42δ =εm .We may repeat the above argument, interchanging the roles of e and 1 −e, todeduce that whenever z ≥0, w ≥0, and z + w < 1 −e,|µ(w + z) −µ(w) −µ(z)| <εm .We now appeal to Lemma 3.2 to obtain|µ(a + b) −µ(a) −µ(b)| < 20ε .Since ε is arbitrary, it follows thatµ(a + b) = µ(a) + µ(b) .Hence µ is linear.When µ is σ-additive, we can give a reasonably straightforward proof of linearityfor Type II1 algebras.

However, in order to obtain Theorem A, it is essential toobtain Theorem B when µ is finitely additive, not positive and not σ-additive.For von Neumann algebras of finite type this forces us to use a more elaborateargument.

6L. J. BUNCE AND J. D. MAITLAND WRIGHT4.

Open problemLet A be a (unital) C∗-algebra. Let φ: A →C be a function whose restrictionto each abelian ∗-subalgebra is linear, is such that {|φ(x)|: ∥x∥≤1} is boundedand whenever x and y are selfadjoint, φ(x + iy) = φ(x) + iφ(y).

Then φ is said tobe a quasi-linear functional.Problem. For which C∗-algebras A is it true that every quasi-linear functional onA is linear?We have the following consequence of Theorem B.Corollary.

Let M be a von Neumann algebra with no direct summand of Type I2.Let I be a closed ideal of M and let A = M/I. Then every quasi-linear functionalon A is linear.Proof.

Let φ: A →C be quasi-linear. Let π: M →M/I be the canonical quotienthomomorphism.

Then the restriction of φπ to P(M) is a finitely additive measurethat, by Theorem B, has a unique extension to a bounded linear functional on M.Hence φ is linear.Corollary. Each quasi-linear functional on the Calkin algebra is linear.References1.

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149 (1970), 601–625.2. L. J. Bunce and J. D. M. Wright, Complex measures on projections in von Neumannalgebras, J. London Math.

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G. W. Mackey, The mathematical foundations of quantum mechanics, Benjamin, 1963.12. S. Maeda, Probability measures on projections in von Neumann algebras, Reviews in Math-ematical Physics 1 (1990), 235–290.13.

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62 (1985), 87–117.14. M. Takesaki, Theory of operator algebras, Springer, 1979.15.

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Soc. 16 (1984), 145–150.16., Measures on projections in W ∗-algebras of Type II1, Bull.

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