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Involutive gradings of JBW-triple factors
Let ( B + , B − ) be an involutive grading of a JBW * -triple factor A with associated involutive triple automorphism φ . When the JBW * -subtriple B + of A is not a JBW * -triple factor there exists a non-zero Peirce weak * -closed inner ideal J in A with Peirce spaces J 0 , J 1 , and J 2 such that...
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| Published in: | Revista matemática complutense 2010-07, Vol.23 (2), p.383-413 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
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| Summary: | Let (
B
+
,
B
−
) be an involutive grading of a JBW
*
-triple factor
A
with associated involutive triple automorphism
φ
. When the JBW
*
-subtriple
B
+
of
A
is not a JBW
*
-triple factor there exists a non-zero Peirce weak
*
-closed inner ideal
J
in
A
with Peirce spaces
J
0
,
J
1
, and
J
2
such that
When both
B
+
and
B
−
are JBW
*
-triple factors it is shown that either the situation reduces to that above with
J
0
or
J
2
equal to zero or, in the case that
B
+
(or, by symmetry,
B
−
) contains a unitary tripotent
v
, that
v
is unitary in
A
, and
where
H
(
A
2
(
v
),
φ
) is the JBW
*
-algebra of
φ
-invariant elements in the JBW
*
-algebra
A
2
(
v
), and
S
(
A
2
(
v
),
φ
) is the JBW
*
-triple of −
φ
-invariant elements of
A
2
(
v
). In the special case in which
A
is a discrete W
*
-factor it is shown that such a unitary tripotent always exists in
B
+
(or
B
−
), thereby completing the description of involutive gradings in this case. |
|---|---|
| ISSN: | 1139-1138 1988-2807 |
| DOI: | 10.1007/s13163-009-0021-z |