In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibits a striking resemblance to the geometry of James space. Further, we show that the averaging projections onto subspaces spanned by constant coefficient blocks with no gaps between supports are bounded. As a consequence, every Banach space with a spreading basis contains a complemented subspace with an unconditional basis. This gives an affirmative answer to a question of H. Rosenthal.
The second part contains two results on Banach spaces
X
X
whose asymptotic structures are closely related to
c
0
c_0
and do not contain a copy of
ℓ
1
\ell _1
:
i) Suppose
X
X
has a normalized weakly null basis
(
x
i
)
(x_i)
and every spreading model
(
e
i
)
(e_i)
of a normalized weakly null block basis satisfies
‖
e
1
−
e
2
‖
=
1
\|e_1-e_2\|=1
. Then some subsequence of
(
x
i
)
(x_i)
is equivalent to the unit vector basis of
c
0
c_0
. This generalizes a similar theorem of Odell and Schlumprecht and yields a new proof of the Elton–Odell theorem on the existence of infinite
(
1
+
ε
)
(1+\varepsilon )
-separated sequences in the unit sphere of an arbitrary infinite dimensional Banach space.
ii) Suppose that all asymptotic models of
X
X
generated by weakly null arrays are equivalent to the unit vector basis of
c
0
c_0
. Then
X
∗
X^*
is separable and
X
X
is asymptotic-
c
0
c_0
with respect to a shrinking basis
(
y
i
)
(y_i)
of
Y
⊇
X
Y\supseteq X
.