C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 2

C O L L O Q U I U M M A T H E M A T I C U M

VOL. 80 1999 NO. 2

MAPPING PROPERTIES OF c 0

BY

PAUL L E W I S (DENTON, TEX.)

Abstract. Bessaga and Pe lczy´ nski showed that if c ₀ embeds in the dual X ^∗ of a Banach space X, then ℓ ¹ embeds as a complemented subspace of X. Pe lczy´ nski proved that every infinite-dimensional closed linear subspace of ℓ ¹ contains a copy of ℓ ¹ that is complemented in ℓ ¹ . Later, Kadec and Pe lczy´ nski proved that every non-reflexive closed linear subspace of L ¹ [0, 1] contains a copy of ℓ ¹ that is complemented in L ¹ [0, 1]. In this note a traditional sliding hump argument is used to establish a simple mapping property of c 0 which simultaneously yields extensions of the preceding theorems as corollaries.

Additional classical mapping properties of c ₀ are briefly discussed and applications are given.

All Banach spaces in this note are defined over the real field. The canon- ical unit vector basis of c ₀ will be denoted by (e _n ), the canonical unit vector basis of ℓ ¹ will be denoted by (e ^∗ _n ), and a continuous linear transformation will be referred to as an operator. The reader is referred to Diestel [3] or Lindenstrauss and Tzafriri [8] for undefined notation and terminology.

Theorem 1. If T : c 0 → X is an operator and (x ^∗ _k ) is any bounded sequence in X ^∗ so that

∞

X

k=1

|x ^∗ _k (T (e _{n k} )) − 1| < ∞

for some subsequence (T (e _{n k} )) of (T (e _n )), then there is a sequence (w ^∗ _i ) in {x ^∗ _k − x ^∗ _j : k, j ∈ N} so that (w ^∗ _i ) is equivalent to (e ^∗ _i ) as a basic sequence and [w ^∗ _i ] is complemented in X ^∗ .

P r o o f. Let (b _k ) = T (e _{n k} ) for k ∈ N, let C be a positive number so that C > 1 and kT (x)k ≤ Ckxk for all x, and choose B > 1 so that

2 sup kx ^∗ _n k < B.

Without loss of generality, suppose that

|x ^∗ _n (b _n ) − 1| < 1 BC · 2 ⁿ⁺⁴

1991 Mathematics Subject Classification: Primary 46B20.

[235]

for each n. Further, since (b _n ) is weakly null, suppose that (1)

n−1

X

i=1

|x ^∗ _i (b n )| < 1 BC · 2 ⁿ⁺⁵

for each n. Now let r 1 = 1, r 2 = 2, and choose r 3 and r 4 so that r 2 < r 3 < r 4

and

|(x ^∗ _{r 4} − x ^∗ _{r 3} )(b _r 2 )| < 1 BC · 2 ¹⁺⁵ . Next choose r 5 and r 6 so that r 4 < r 5 < r 6 and

(2)

|(x ^∗ _{r 6} − x ^∗ _{r 5} )(b _r 2 )| < 1 BC · 2 ²⁺⁵ ,

|(x ^∗ _{r 6} − x ^∗ _{r 5} )(b _r 4 )| < 1 BC · 2 ²⁺⁵ .

An additional step clarifies the induction process. Choose r ₇ and r ₈ so that r 6 < r 7 < r 8 and

(3)

|(x ^∗ _{r 8} − x ^∗ _{r 7} )(b _r 2 )| < 1 BC · 2 ³⁺⁵ ,

|(x ^∗ _{r 8} − x ^∗ _{r 7} )(b _r 4 )| < 1 BC · 2 ³⁺⁵ ,

|(x ^∗ _{r 8} − x ^∗ _{r 7} )(b r 6 )| < 1 BC · 2 ³⁺⁵ .

Continue this construction inductively, and let u _n = b _{r 2n} and z _n ^∗ = x ^∗ _{r 2n} for each n. Note that

|z ^∗ _i (u _n ) − x ^∗ _{r 2}

i−1 (u _n )| < 1 BC · 2 ⁱ⁺⁴ for n < i. Further,

|z _n ^∗ (u _n ) − 1| < 1

BC · 2 ⁿ⁺⁴ and

n

X

i=1

|z ^∗ _i (u n+1 )| < 1 BC · 2 ^(n+1)+5 for each n.

Next let w ^∗ _n = z _n ^∗ − x ^∗ _{r 2 n−1} = x ^∗ _{r 2n} − x ^∗ _{r 2 n−1} for n ∈ N. Then

|w _n ^∗ (u _n ) − 1| ≤ |z _n ^∗ (u _n ) − 1| + |x ^∗ _{r 2 n−1} (u _n )|

< 1

BC · 2 ⁿ⁺⁴ + 1

BC · 2 ⁿ⁺⁵ = 3 BC · 1

2 ⁿ⁺¹ · 1 2 ⁴ . Also, kx ^∗ _n k ≤ B for each n.

Now suppose that q ∈ N and t _i is a non-zero real number for 1 ≤ i ≤ q.

If ε _i = sgn(t _i w _i ^∗ (u _i )), then

q

X

i=1

t i w ^∗ _i (ε i u 1 ) ≥ |t 1 w ^∗ ₁ (u 1 )| −

q

X

i=2

|w _i ^∗ t i (u 1 )|

≥ |t 1 |

 1 − 3

BC · 1 2 ² · 1

2 ⁴



−

 |t 2 |

BC · 2 ¹⁺⁵ + . . . + |t q | BC · 2 ^(q−1)+5

 . Further,

q

X

i=1

t _i w ^∗ _i (ε 2 u 2 )

= |t 2 w ^∗ ₂ (u 2 )| −

q

X

i=1,i6=2

t _i w ^∗ _i (ε 2 u 2 )

≥ |t 2 |

 1 − 3

BC · 1 2 ³ · 1

2 ⁴



−



|t 1 | 1 BC · 1

2 ²⁺⁵ + |t 3 | 1 BC · 1

2 ²⁺⁵ + . . . + |t q | 1

BC · 1

2 ^(q−1)+5

 . (Observe that

|w ^∗ ₁ (u ₂ )| = |(x ^∗ ₂ − x ^∗ ₁ )(b _r 4 )| ≤ |x ^∗ ₂ (b _r 4 )| + |x ^∗ ₁ (b _r 4 )|

< 1

BC · 2 ^{r 4 +5} + 1

BC · 2 ^{r 4 +5} < 1 BC · 2 ²⁺⁵ from (1). Also,

|w ^∗ ₃ (u 2 )| = |(x ^∗ _{r 6} − x ^∗ _{r 5} )(b _r 4 )| ≤ 1 BC · 2 ²⁺⁵ from (2), and

|(x ^∗ _{r 8} − x ^∗ _{r 7} )(b _r 4 )| < 1 BC · 2 ³⁺⁵ from (3).)

In general, D X ^q

i=1

ε _i u _i ,

q

X

n=1

t _n w ^∗ _n E

≥ |t ₁ |

 1 − 3

BC · 1 2 ² · 1

2 ⁴



− |t 1 |

 1

BC · 2 ^{r 4 +5} + 1

BC · 2 ^{r 6 +5} + . . . + 1 BC · 2 ^{r 2q +5}



+ |t 2 |

 1 − 3

BC · 1 2 ³ · 1

2 ⁴



− |t 2 |

 1

BC · 2 ²⁺⁴ + 1

BC · 2 ^{r 6 +5} + . . . + 1 BC · 2 ^{r 2q +5}



+ |t ₃ |

 1 − 3

BC · 1 2 ⁴ · 1

2 ⁴



− |t 3 |



2 1

BC · 2 ³⁺⁴ + 1

BC · 2 ^{r 8 +5} + 1

BC · 2 ^{r 10 +5} + . . . + 1 BC · 2 ^{r 2q +5}



+ |t 4 |

 1 − 3

BC · 1 2 ⁵ · 1

2 ⁴



− |t ₄ |



3 1

BC · 2 ⁴⁺⁴ + 1

BC · 2 ^{r 10 +5} + . . . + 1 BC · 2 ^{r 2q +5}



+ . . . +

+ |t _q |

 1 − 3

BC · 1 2 ^q+1 · 1

2 ⁴



− |t _q |

 q − 1 BC · 2 ^q+4

 . Note that

3

BC · 2 ² 2 ⁴ + 1

BC · 2 ^{r 4 +5} + . . . + 1

BC · 2 ^{r 2q} ≤ 2 BC · 2 ⁴ , 3

BC · 2 ³ 2 ⁴ + 1

BC · 2 ²⁺⁴ + 1

BC · 2 ^{r 6 +5} + . . . + 1

BC · 2 ^{r 2q +5} ≤ 2 BC · 2 ⁴ , . . .

3

BC · 2 ^q+1 2 ⁴ + q − 1

BC · 2 ^q+4 ≤ 2 BC · 2 ⁴ . Consequently,

D X ^q

i=1

ε _i u _i ,

q

X

n=1

t _n w ^∗ _n E

≥  X ^q

i=1

|t _i |  

1 − 2

BC · 2 ⁴



> 0.

Thus P q

i=1 ε _i u _i 6= 0, and

q

X

i=1

t _i w ^∗ _i ≥

 1 .

q

X

i=1

ε _i u _i

D X ^q

i=1

ε _i u _i ,

q

X

n=1

t _n w _n ^∗ E

≥  X ^q

i=1

|t _i |  

1 − 1

BC · 2 ³

 c ⁻¹

 . Hence



1 − 1

BC · 2 ³

 c ⁻¹

  X ^q

i=1

|t _i | 

≤

q

X

i=1

t _i w ^∗ _i ≤ B

q

X

i=1

|t _i |, and (w ^∗ _i ) ∼ (e ^∗ _i ).

Next we show that [w _n ^∗ ] is complemented in X ^∗ . Suppose that v ^∗ = P ∞

n=1 t _n w ^∗ _n , and let U : X ^∗ → [w ^∗ _n ] be defined by

U (x ^∗ ) = X

n

x ^∗ (u _n )w ^∗ _n .

Since (u n ) is a subsequence of (b k ) and P b k is weakly unconditionally con- vergent, it is clear that U is well defined, continuous, and linear. Now observe that

kv ^∗ − U (v ^∗ )k

=

X t n w _n ^∗ − X

t n U (w _n ^∗ )

=

∞

X

n=1

t n w _n ^∗ −

∞

X

n=1

t n

 X ^∞

k=1

w ^∗ _n (u k )w ^∗ _k 

=

∞

X

n=1

t n w _n ^∗ −

∞

X

n=1

t n w ^∗ _n (u n )w ^∗ _n −

∞

X

n=1

t n

 X ^∞

k=1, k6=n

w ^∗ _n (u k )w ^∗ _k 

≤

∞

X

n=1

|t n | · |1 − w ^∗ _n (u n )| · kw ^∗ _n k +

∞

X

n=1

|t n |  X ^∞

k=1, k6=n

|w ^∗ _n (u k )|B 

≤

∞

X

n=1

|t n |  sup

k

n

|1 − w ^∗ _k (u k )| +

∞

X

i=1, i6=k

|w ^∗ _k (u i )| o

B.

Also,

∞

X

n=1

|t n | ≤ c 1 − _BC·2 ¹ 3

kv ^∗ k.

Further,

sup

k

|1 − w _k ^∗ (u _k )| ≤ 3 BC · 2 ² 2 ⁴ , and kw ^∗ _k k ≤ B for each k.

Next note that

∞

X

k=2

|w ^∗ ₁ (u k )| =

∞

X

k=2

|(x ^∗ ₂ − x ^∗ ₁ )(u k )|

= |(x ^∗ ₂ − x ^∗ ₁ )(T (e _r 4 ))| + |(x ^∗ ₂ − x ^∗ ₁ )(T (e _r 6 ))| + . . .

≤ (|x ^∗ ₂ T (e _r 4 )| + |x ^∗ ₁ T (e _r 4 )|) + (|x ^∗ ₂ T (e _r 6 )| + |x ^∗ ₁ T (e _r 6 )|) + . . .

< 1

BC · 2 ^{r 4 +5} + 1

BC · 2 ^{r 6 +5} + . . . < 1

BC · 2 ^{r 4 +4} < 1 BC · 2 ⁴ . A similar argument shows that

∞

X

i=1, i6=k

|w ^∗ _k (u _i )| < 1

BC · 2 ⁴

for each k. Thus

kv ^∗ − U (v ^∗ )k ≤ c 1 − _BC·2 ¹ 3

kv ^∗ k

 3

BC · 2 ² 2 ⁴ + 1 BC · 2 ⁴

 B < 1

7 kv ^∗ k.

If U 1 = U _|[w ∗

i ] , then kIdentity − U 1 k _|[w ∗

i ] < 1, and U 1 is invertible on [w ^∗ _i ]. It is easy to see that U ₁ ⁻¹ U is a projection from X ^∗ onto [w ^∗ _i ].

Remark . (a) The operator T : c 0 → X satisfies the hypotheses of Theorem 1 if and only if lim inf kT (e _n )k > 0. H. Rosenthal [11] has given a penetrating study of the situation in which T : ℓ ^∞ (Γ ) → X is an operator so that inf γ∈Γ kT (e γ )k > 0.

(b) If (x ^∗ _k ) is w ^∗ -null, the proof of Theorem 1 makes it clear that we may choose the sequence (w ^∗ _i ) in the conclusion of the theorem to be w ^∗ -null.

As the following corollaries indicate, Theorem 1 unifies and extends sev- eral classical results.

Corollary 2 ([1, Thm. 4], [3, p. 48]). If c ₀ embeds isomorphically in the dual X ^∗ of the Banach space X, then X contains a copy of ℓ ¹ which is complemented (in X ^∗∗ and thus) in X.

P r o o f. If T : c 0 → X ^∗ is an isomorphism, then let (x _n ) be a bounded sequence in X (⊆ X ^∗∗ ) so that P ∞

n=1 |x n (T (e n ))−1| = 0. Apply Theorem 1 to the sequence (x n ).

Corollary 3 ([10], [3, p. 72]). If ℓ ¹ is a quotient of X, then X contains a copy of ℓ ¹ which is complemented in X ^∗∗ .

P r o o f. If T : X → ℓ ¹ is a surjective operator, then T ^∗ : ℓ ^∞ → X ^∗ is an isomorphism. Hence T _|c ^∗

0 is an isomorphism.

If Σ is a σ-algebra, (µ _n ) is a bounded sequence in cabv(Σ, X), and 0 < ε < δ, then (µ n ) is said to be (δ, ε)-relatively disjoint [11] if there is a pairwise disjoint sequence (A _n ) in Σ so that

|µ _n |(A _n ) > δ and

∞

X

m=1, m6=n

|µ _n |(A _m ) < ε

for each n. Further, (µ _n ) is said to be relatively disjoint if it is (δ, ε)-relatively disjoint for some pair (δ, ε). Rosenthal [11] and Kadec and Pe lczy´ nski [7]

showed that if (µ n ) is a relatively disjoint sequence in cabv(Σ, X), then (µ _n ) ∼ (e ^∗ _n ) and [µ _n ] is complemented in cabv(Σ, X).

If A is an algebra of subsets of Ω, then fabv(A, X) denotes the Banach

space (total variation norm) of all finitely additive set functions m : A → X

which have finite variation. Both [4] and [6] contain an extensive discussion

of spaces of measures. In addition, we note that [4] includes a detailed

presentation of results related to the Radon–Nikodym property. Note that

part (i) of Corollary 4 below contains an extension of Proposition 3.1 of [11]

to the setting of finitely additive set functions defined on an algebra of sets.

Further, we remark that in a classic paper Kadec and Pe lczy´ nski [7, Theo- rem 6] showed that if Y is any non-reflexive closed linear subspace of L ¹ [0, 1], then Y contains a copy of ℓ ¹ which is complemented in L ¹ [0, 1]. Part (v) of the next corollary shows that if X and X ^∗ have the Radon–Nikodym property, then any non-reflexive closed linear subspace of L ¹ (µ, X) contains a copy of ℓ ¹ which is complemented in L ¹ (µ, X).

Corollary 4. (i) If (µ n ) is any bounded sequence in fabv(A, X) for which there is a pairwise disjoint sequence (A _n ) in A and an ε > 0 so that

|µ n |(A n ) > ε

for each n, then there is a sequence (ν _i ) in {µ _n − µ _k : k, n ∈ N} so that (ν i ) ∼ (e ^∗ _i ) and [ν i ] is complemented in fabv(A, X).

(ii) If K is a relatively weakly compact subset of fabv(A, X) and (A _i ) is a pairwise disjoint sequence of members of A, then lim i |µ|(A i ) = 0 uni- formly for µ ∈ K.

(iii) If K is a relatively weakly compact subset of cabv(Σ, X), then {|µ| : µ ∈ K} is uniformly countably additive.

(iv) If µ is a finite positive measure on Σ and K is a relatively weakly compact subset of the space L ¹ (µ, X) of Bochner integrable functions, then K is uniformly integrable.

(v) If Y is a closed linear subspace of fabv(A, X), Y is not reflexive, and X and X ^∗ have the Radon–Nikodym property, then Y contains a copy of ℓ ¹ which is complemented in fabv(A, X).

P r o o f. (i) For each n let (A _{n i} ) ^k _i=1 ⁿ be a partition of A _n and (x ^∗ _{n i} ) ^k _i=1 ⁿ be points in the unit ball of X ^∗ so that

k n

X

i=1

x ^∗ _{n i} µ n (A n i ) > ε.

Now define the X ^∗ -valued simple function s _n by s _n =

k n

X

i=1

χ _{A ni} x ^∗ _{n i} , and observe that

T

s n dµ n > ε. Define T : c 0 → fabv(A, X) ^∗ by T ((γ n )) = X

n

γ n s n .

Then T is an operator. Normalize and use Theorem 1 to conclude that some

sequence (ν _i ) in {µ _n − µ _k : n, k ∈ N} is equivalent to (e ^∗ _n ) and that [ν _n ] is

complemented in fabv(A, X).

(ii) Suppose that ε > 0 and (µ _i ) is a sequence in K so that |µ _i |(A _i ) > ε for each i. Part (i) ensures that (e ^∗ _n ) is equivalent to some sequence in K −K.

However, this is impossible since K − K is relatively weakly compact.

(iii) Since each member of K is a countably additive measure on a σ-algebra, |K| = {|µ| : µ ∈ K} is uniformly countably additive if and only if lim i |µ|(A i ) = 0 uniformly for µ ∈ K whenever (A i ) is a pairwise disjoint sequence from Σ. Deny the uniform countable additivity of |K|, repeat the same construction as in (i), and obtain the same contradiction as in (ii).

(iv) If f ∈ L ¹ (µ, X) and A ∈ Σ, put ν _f (A) =

\

A

f dµ.

It is well known that lim µ(A)→0 |ν f |(A) = 0 uniformly for f ∈ K (i.e., K is uniformly integrable) if and only if {|ν _f | : f ∈ K} is uniformly countably additive. Appeal to (iii).

(v) If Y is not reflexive, then B Y is not relatively weakly compact in fabv(A,X). By Theorem 4.1 of Brooks and Dinculeanu [2], there is a pairwise disjoint sequence (A i ) in A, an ε > 0, and a sequence (µ i ) in B Y so that

|µ i |(A i ) > ε for each i. The construction in (i) above shows that Y contains a copy of ℓ ¹ which is complemented in fabv(A, X).

In the following corollary, P denotes the σ-algebra of all subsets of N.

Corollary 5 ([9, Lemma 2], [3, p. 74]). Every infinite-dimensional closed linear subspace of ℓ ¹ contains a copy of ℓ ¹ which is complemented in fabv(P) and thus in ℓ ¹ .

P r o o f. Every infinite-dimensional subspace of ℓ ¹ is non-reflexive.

Corollary 6 ([4, p. 149]). If (Ω, Σ, µ) is a finite measure space and X ^∗ is a quotient of L ^∞ (µ), then either X is reflexive or X contains a copy of ℓ ¹ which is complemented in X ^∗∗ . Consequently , if X ^∗∗ is contained in L ¹ (µ), then X is reflexive or ℓ ¹ is a complemented subspace of X.

P r o o f. If T : L ^∞ (µ) → X ^∗ is a surjection and X is not reflexive, then T is not weakly compact. Hence T is not unconditionally converging and is an isomorphism on a copy of c 0 . Thus X contains a copy of ℓ ¹ which is complemented in X ^∗∗ .

If L : X ^∗∗ → L ¹ (µ) is an isomorphism, then L ^∗ : L ^∞ (µ) → X ^∗∗∗ is a surjection, X ^∗ is a quotient of L ^∞ (µ), and X is reflexive or X contains a complemented copy of ℓ ¹ .

If T : c 0 → X is an isomorphism, classical techniques of Singer [13] can be used to easily produce complemented copies of both c 0 and ℓ ¹ .

Theorem 7. If T : c ₀ → X is an isomorphism, (f _n ^∗ ) is any bounded

sequence in X ^∗ so that

f _n ^∗ (T (e _m )) = δ _nm ,

and (h ^∗ _k ) is any subsequence of (f _n ^∗ ), then [h ^∗ _k ] is complemented in X ^∗ . Further , if (h ^∗ _k ) is w ^∗ -null in X ^∗ and (y _k ) is the corresponding subsequence of (T (e n )), then [y k ] is complemented in X.

P r o o f. Suppose that T, (f _n ^∗ ), and (h ^∗ _k ) are as in the first statement in the theorem. Let C be a bound for (kf _n ^∗ k), let (y ^∗ _k ) be the sequence of coefficient functionals for the basic sequence (y _k ) (which is equivalent to (e _k )), and choose positive numbers A and B so that

A X

|α _i | ≤

X α _i y ^∗ _i ≤ B

X |α _i |

for each finite sequence (α 1 , . . . , α _m ) of real numbers. Therefore

A X

|α _i | ≤

X α _i h ^∗ _{i|[y n ]} ≤

X α _i h ^∗ _i ≤ C

X |α _i |.

As noted on p. 91 of Singer [13], n

f ^∗ ∈ X ^∗ :

∞

X

k=1

f ^∗ (y _k )h ^∗ _k converges o

= [y _k ] ^⊥ + [h ^∗ _k ].

Since (y _k ) ∼ (e _k ) and (h ^∗ _k ) ∼ (e ^∗ _k ), we have [y _k ] ^⊥ + [h ^∗ _k ] = X ^∗ . Further, if (h ^∗ _k ) is w ^∗ -null, then

n x ∈ X :

∞

X

k=1

h ^∗ _k (x)y _k converges o

= [y _k ] + [h ^∗ _k ] ⊥ = X.

Consequently, each of these direct sums is closed. Straightforward closed graph arguments show that these direct sums are also topological.

We remark that if X is separable (and T and (f _n ^∗ ) have the same meaning as in the statement of Theorem 7), then Veech’s proof [15] of Sobczyk’s theorem [14], [3, p. 71] simply shows that there is a bounded sequence (g ^∗ _n ) in [T (e _n ))] ^⊥ so that (f _n ^∗ −g _n ^∗ ) is w ^∗ -null. Certainly (T (e _n ), f _n ^∗ −g ^∗ _n ) is biorthog- onal in this case.

The next corollary shows that a result of Saab and Saab [12] dealing with complemented copies of c 0 in injective tensor products is an immediate consequence of Theorem 7. Chapter 8 of [4] contains an excellent discussion of the least crossnorm tensor product completion of Banach spaces.

Corollary 8 ([12]). If X contains a copy of c ₀ , Y is an infinite-dimen- sional Banach space and Z = X ⊗ _λ Y is the least crossnorm tensor product completion of X and Y , then Z contains a complemented copy of c 0 .

P r o o f. Let (x _n ) be a sequence in X so that (x _n ) ∼ (e _n ), let (x ^∗ _n ) be

a bounded sequence in X ^∗ so that x ^∗ (x m ) = δ nm , and let (y _n ^∗ ) be a w ^∗ -null

sequence in Y ^∗ so that ky ^∗ _n k = 1 for each n. (The Josefson–Nissenzweig

Theorem [3] guarantees the existence of (y _n ^∗ ).) Choose a sequence (y _n ) in Y

so that ky _n k ≤ 3/2 and y _n (y _n ^∗ ) = 1 for each n. Then (x ^∗ _n ⊗ y _n ^∗ ) is a w ^∗ -null sequence in Z ^∗ , (x _n ⊗ y _n ) ∼ (e _n ), and x ^∗ _n ⊗ y ^∗ _n (x _m ⊗ y _m ) = x ^∗ _n (x _n )y _n ^∗ (y _m )

= δ _nm . Now appeal to Theorem 7.

We note that precisely the same argument yields the next result.

Corollary 9. If the Banach space X contains a copy of c 0 and Y is an infinite-dimensional space , then the Banach space K(X ^∗ , Y ) of compact operators from X ^∗ to Y contains a complemented copy of c 0 .

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University of North Texas Denton, Texas

E-mail: lewis@unt.edu

Received 10 November 1998