Author:
Ka̧kol Jerzy,Śliwa Wiesław
Abstract
AbstractLet $${\mathbb {K}}$$
K
be a non-trivially valued non-Archimedean complete field. Let $$\ell _{\infty }({\mathbb {N}}, {\mathbb {K}})$$
ℓ
∞
(
N
,
K
)
[$$\ell _c({\mathbb {N}}, {\mathbb {K}});$$
ℓ
c
(
N
,
K
)
;
$$c_0({\mathbb {N}}, {\mathbb {K}})$$
c
0
(
N
,
K
)
] be the space of all sequences in $${\mathbb {K}}$$
K
that are bounded [relatively compact; convergent to 0] with the topology of pointwise convergence (i.e. with the topology induced from $${\mathbb {K}}^{{\mathbb {N}}}$$
K
N
). Let X be an infinite ultraregular space and let $$C_p(X,{\mathbb {K}})$$
C
p
(
X
,
K
)
be the space of all continuous functions from X to $${\mathbb {K}}$$
K
endowed with the topology of pointwise convergence. It is easy to see that $$C_p(X,{\mathbb {K}})$$
C
p
(
X
,
K
)
is metrizable if and only if X is countable. We show that for any X [with an infinite compact subset] the space $$C_p(X,{\mathbb {K}})$$
C
p
(
X
,
K
)
has an infinite-dimensional [closed] metrizable subspace isomorphic to $$c_0({\mathbb {N}}, {\mathbb {K}})$$
c
0
(
N
,
K
)
. Next we prove that $$C_p(X,{\mathbb {K}})$$
C
p
(
X
,
K
)
has a quotient isomorphic to $$c_0({\mathbb {N}}, {\mathbb {K}})$$
c
0
(
N
,
K
)
if and only if it has a complemented subspace isomorphic to $$c_0({\mathbb {N}}, {\mathbb {K}})$$
c
0
(
N
,
K
)
. It follows that for any extremally disconnected compact space X the space $$C_p(X,{\mathbb {K}})$$
C
p
(
X
,
K
)
has no quotient isomorphic to the space $$c_0({\mathbb {N}}, {\mathbb {K}})$$
c
0
(
N
,
K
)
; in particular, for any infinite discrete space D the space $$C_p(\beta D, {\mathbb {K}})$$
C
p
(
β
D
,
K
)
has no quotient isomorphic $$c_0({\mathbb {N}}, {\mathbb {K}})$$
c
0
(
N
,
K
)
. Finally we investigate the question for which X the space $$C_p(X,{\mathbb {K}})$$
C
p
(
X
,
K
)
has an infinite-dimensional metrizable quotient. We show that for any infinite discrete space D the space $$C_p(\beta D, {\mathbb {K}})$$
C
p
(
β
D
,
K
)
has an infinite-dimensional metrizable quotient isomorphic to some subspace $$\ell _c^0({\mathbb {N}}, {\mathbb {K}})$$
ℓ
c
0
(
N
,
K
)
of $${\mathbb {K}}^{{\mathbb {N}}}$$
K
N
. If $${\mathbb {K}}$$
K
is locally compact then $$\ell _c^0({\mathbb {N}}, {\mathbb {K}})=\ell _{\infty }({\mathbb {N}}, {\mathbb {K}})$$
ℓ
c
0
(
N
,
K
)
=
ℓ
∞
(
N
,
K
)
. If $$|n1_{{\mathbb {K}}}|\ne 1$$
|
n
1
K
|
≠
1
for some $$n\in {\mathbb {N}}$$
n
∈
N
, then $$\ell _c^0({\mathbb {N}}, {\mathbb {K}})=\ell _c ({\mathbb {N}}, {\mathbb {K}}).$$
ℓ
c
0
(
N
,
K
)
=
ℓ
c
(
N
,
K
)
.
In particular, $$C_p(\beta D, {\mathbb {Q}}_q)$$
C
p
(
β
D
,
Q
q
)
has a quotient isomorphic to $$\ell _{\infty }({\mathbb {N}}, {\mathbb {Q}}_q)$$
ℓ
∞
(
N
,
Q
q
)
and $$C_p(\beta D, {\mathbb {C}}_q)$$
C
p
(
β
D
,
C
q
)
has a quotient isomorphic to $$\ell _c({\mathbb {N}}, {\mathbb {C}}_q)$$
ℓ
c
(
N
,
C
q
)
for any prime number q.
Funder
Ministerio de Ciencia, Innovación y Universidades
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Geometry and Topology,Algebra and Number Theory,Analysis
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