Maps between Banach algebras preserving the spectrum
Let A and B be unital semisimple complex Banach algebras, and let φ 1 and φ 2 be maps from A onto B . We show that if the socle of A is an essential ideal of A , and φ 1 and φ 2 satisfy σ ( φ 1 ( a ) φ 2 ( b ) ) = σ ( a b ) for all a , b ∈ A , then φ 1 φ 2 ( 1 ) and φ 1 ( 1 ) φ 2 coincide and are Jo...
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| Veröffentlicht in: | Archiv der Mathematik 2016-12, Vol.107 (6), p.609-621 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | Let
A
and
B
be unital semisimple complex Banach algebras, and let
φ
1
and
φ
2
be maps from
A
onto
B
. We show that if the socle of
A
is an essential ideal of
A
, and
φ
1
and
φ
2
satisfy
σ
(
φ
1
(
a
)
φ
2
(
b
)
)
=
σ
(
a
b
)
for all
a
,
b
∈
A
, then
φ
1
φ
2
(
1
)
and
φ
1
(
1
)
φ
2
coincide and are Jordan isomorphisms. We also show that a map
φ
from
A
onto
B
satisfies
σ
(
φ
(
a
)
φ
(
b
)
φ
(
a
)
)
=
σ
(
a
b
a
)
for all
a
,
b
∈
A
if and only if
φ
(
1
)
is a central invertible element of
B
for which
φ
(
1
)
3
=
1
and
φ
(
1
)
2
φ
is a Jordan isomorphism. |
|---|---|
| ISSN: | 0003-889X 1420-8938 |
| DOI: | 10.1007/s00013-016-0960-9 |