Periodica Mathematica Hungarica Vol. 9 (4), (1978), pp. 293--295

B O O L E A N A L G E B R A S O F P R O J E C T I O N S

by

1%. EVANS (Berlin)

A family A of commuting projections on a Banach space X such that (i) I , 0 ~ A, (ii) E ~ A ~ 1 -- E E A, (iii) E, F E A ~ B F E A forms a Boolean algebra with the lattice operations E V .F ~ E ~ F -- EF , E A F ~ E F . I f m is a a-finite measure, the family of all characteristic projections (i.e., / -~ ZDf for m-measurable D) on LP(m) is a simple example of such a Boolean algebra. These projections have the property that for all x,

ilxll p = IIExll p + I I ( I - - E ) x l I p, 1 < p ~ + or

l[xll = max {lIEx(I, [ 1 ( ! - E)xl[} , p = + ~ .

In the general setting, projections with this property for some p are called LP-projections. The purpose of this paper is to prove the following theorem, which shows that a relatively weak condition on a generM Boolean algebra implies that it consists solely of LP-projections.

THEOREM. Let X be a Banach space and A a Boolean algebra of projec- tions on X with I A I ~ 4. I f for every E(E A) and x, y (in X )

liExil > IIEy]l and I i } ( i _ E ) x ] ] > ] ] ( i _ F ~ ) y ] ] , ~ [Ix[I > l[Yl[,

then A consist8 solely of LP-pro]ections for some fixed p, 1 ~ p ~ ~- r

PROOF. The proof is broken down into the following steps:

(a) for every E ( E A ) there is a function JR: R + • such that l i x { I : fE(lIExll, l l ( I - - E ) x l l ) for all x in X.

(b) for every pair E, F(E A), 0 ~ E ~ I , 0 ~ F ~ I , f~ = fF ( = some fixed f).

A]IS (MOS) subject classifications (1970). Primary 47D99; Secondary 46L20. Key words and phrases. LP-projection, :Boolean algebra.

294 EVANS: BOOLEAN ALGEBRA8 OF FROJEOTIONS

(c) f has the form / ( r , s ) : ( r p+sp) lip for some l ~ p < + o o or

t(*, ~) = ma~ {~, ~}.

PROOF of (a). I f E ~ I or E ~ 0, then any funct ion such tha t f(r, O) = f(0, r) ~ r will do. I f E # I and E # 0, let

xEEx , y E ( I - - E ) X , ilxil : ][Yil = 1.

Define ]~: R+ X R+ -* R+ by

/~(r, s): = I I rx + sy l l , r, s E It+.

Suppose there is an element z in X such t h a t

iizll ~ / ~ . ( l l E z t l , I I ( I - B)z l l ) .

Without loss of general i ty lot

Ilzlt > f~( l tEzl i , II(I - - E)z i t ) .

Then there is an e ( > 0) such t h a t

(1 - e)llzil > f s ( l lEz t l , l i (I-E)zl l) .

We write r for ]J Ezl[ ~nd 8 for ]l (I -- E) z ll. Clearly both r and s are non-zero, since, if r is zero, 8 is I I z I] and the equal i ty is tr ivial (and vice versa). But , nOW

and thus

I]rxlJ ~ I[E(1--~)z[],

ItsYII ~ I](I--E)(1 e)zll,

Ilrx + sy]] > ]l(1-- e)zll,

which contradicts the choico of z and e.

PROOF of (b). Le t E, F E A wi th 0 ~ E ~ F ~ I . Le t x E E X ~ F X and y E (I - F) X ~ (I -- E) X wi th II x I[ -~ I I y I] -~ 1. Then for every, r, s in R+,

Mr, 8) = I lrz + 8yl] --It(r, s),

i.e., f• ~ IF- Now suppose t h a t E and F are in A with 0 ~ E, F ~ I and E ~ I -- F . Then ei ther E A F s~ 0 or E V F =z~ I , say the former. B y wha t we have jus t shown, f~AF-~fE and also fE^F-----/~, i.e., ] ~ : ] ~ . I t remains only to show t h a t fE-----f1-E, bu t as ] A I ~ 4, there is a th i rd non-tr ivial projection F in A and then f~ -~ fF ---- f1-E.

PROOF of (c). Since I A I ~ 4, there are two projections E, F in A with 0 ~ E ~ F ~ I . L e t x E E X , y E ( I - -B ) FX, z E ( I - - F ) X w i t h ltx[I -~ Ily[I :

EVANS: BOOLEAN ALGEBRAS OF FP~OJECTIONS 295

= [J z[I ~ 1. Let f be the function associated with the projections in A. The following properties of f are trivial:

(i) /(rs, r t ) = rf(s, t) for all r, s, t in R+. (ii) I(1, 0) = 1.

(iii) f(a, t ) = f(t, 8) for all s, t in R+.

I t follows also directly from our condition on A that

(iv) /(a, b) K f(r, a) for a • r, b K 8.

Let r, s, t be in R+ and consider the element rx ~- 8y + tz in X. Decompos- ing first with F and then with E we get

II rx + sy + tz I I = / ( l l rx + ~J]l, t) = / i f ( r , 8), 0-

On the other hand, a decomposition with E first and then F gives

II rx + sy q-- tz[I = f(r , Ilsy + tz l l ) = f (r , f(8, t)).

We thus have

(v) l ( l ( r , 8), t) = l ( r , I(8, t)) for all r, s, t in R+.

:BOHNENBLUST ([1], Theorem 4.1) has shown that a function with these five properties is either max {r, 8} or has the form f(r, s ) = (r p + sP) I/p for some p( ~ 0). The triangle inequality for the norm in a Banach space restricts us in our case to those p ~ 1.

R E F E R E N C E

[1] F . BOHNENBLUST, A n axioma$ie cha rac te r i sa t ion o f LP-spaces, Duke Math. J. 6 (1940), 627--640. M R 2- -102

(Received May 14, 1976)

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