%0 Electronic Article
%A Conway, John B.
%I American Mathematical Society (AMS)
%D 1975
%D 1975
%G English
%@ 0002-9947
%@ 1088-6850
%~ Universitätsbibliothek "Georgius Agricola"
%T On the Calkin algebra and the covering homotopy property
%V 211
%J Transactions of the American Mathematical Society
%V 211
%P 135-142
%U http://dx.doi.org/10.1090/s0002-9947-1975-0399875-4
%X
Let
H
\mathcal {H}
be a separable Hilbert space,
B
(
H
)
\mathcal {B}(\mathcal {H})
the bounded operators on
H
,
K
\mathcal {H},\mathcal {K}
the ideal of compact operators, and
π
\pi
the natural map from
B
(
H
)
\mathcal {B}(\mathcal {H})
onto the Calkin algebra
B
(
H
)
/
K
\mathcal {B}(\mathcal {H})/\mathcal {K}
. Suppose X is a compact metric space and
Φ
:
C
(
X
)
×
[
0
,
1
]
→
B
(
H
)
/
K
\Phi :C(X) \times [0,1] \to \mathcal {B}(\mathcal {H})/\mathcal {K}
is a continuous function such that
Φ
(
⋅
,
t
)
\Phi ( \cdot ,t)
is a
∗
\ast
-isomorphism for each t and such that there is a
∗
\ast
-isomorphism
ψ
:
C
(
X
)
→
B
(
H
)
\psi :C(X) \to \mathcal {B}(\mathcal {H})
with
π
ψ
(
⋅
)
=
Φ
(
⋅
,
0
)
\pi \psi ( \cdot ) = \Phi ( \cdot ,0)
. It is shown in this paper that if X is a simple Jordan curve, a simple closed Jordan curve, or a totally disconnected metric space then there is a continuous map
Ψ
:
C
(
X
)
×
[
0
,
1
]
→
B
(
H
)
\Psi :C(X) \times [0,1] \to \mathcal {B}(\mathcal {H})
such that
π
Ψ
=
Φ
\pi \Psi = \Phi
and
Ψ
(
⋅
,
0
)
=
ψ
(
⋅
)
\Psi ( \cdot ,0) = \psi ( \cdot )
. Furthermore if X is the disjoint union of two spaces that both have this property, then X itself has this property.
%Z https://katalog.ub.tu-freiberg.de/Record/ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAwMi05OTQ3LTE5NzUtMDM5OTg3NS00
%U https://katalog.ub.tu-freiberg.de/Record/ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAwMi05OTQ3LTE5NzUtMDM5OTg3NS00