Measures on the Lattice of Closed Inner Ideals in a Spin Triple

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 thatWhen 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, andwhere 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.

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“…Its solution in various cases is given in [18,19,[25][26][27]. However, when case (i) occurs it is less clear how further progress can be made.…”

Section: Introductionmentioning

confidence: 99%

Involutive gradings of JBW*-triple factors

Edwards

Morton

2009

Rev Mat Complut

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“…On the other hand, when A is a spin triple or one of the two exceptional Cartan factors, most of the weak * -closed inner ideals in which are not Peirce, the result shows that such inner ideals cannot occur as the constituents J 0 and J 2 of a Peirce grading (J 0 , J 1 , J 2 ). The reader is referred to [11,13,14,15,16,19] for details. is said to be a Jordan * -triple.…”

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confidence: 99%

Peirce gradings and Peirce inner ideals in JBW*-triple factors

Edwards

Morton

2008

Arch. Math.

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A Peirce inner ideal J in an anisotropic Jordan * -triple A gives rise to a Peirce grading (J0, J1, J2) of A by defining J0 = J ⊥ , J1 = Ker(J ) ∩ Ker(J ⊥ ), J2 = J, where J ⊥ is the set of elements a of A for which {J a A} is equal to {0} and Ker(J ) is the set of elements a of A for which {J a J} is equal to {0}. It is shown that conversely, when A is a JBW * -triple factor, for each Peirce grading (J0, J1, J2) of A such that both J0 and J2 are non-zero, both J0 and J2 are Peirce inner ideals the corresponding Peirce decompositions of A being given by (J0)0 = J2, (J0)1 = J1, (J0)2 = J0; (J2)0 = J0, (J2)1 = J1, (J2)2 = J2. Mathematics Subject Classification (2000). Primary 17C65; Secondary 46L70.

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Complementary inner ideals in JBW⁎-triples

Lubbe

2012

Journal of Algebra

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Measures on the Lattice of Closed Inner Ideals in a Spin Triple

Cited by 3 publications

References 31 publications

Involutive gradings of JBW*-triple factors

Involutive gradings of JBW*-triple factors

Peirce gradings and Peirce inner ideals in JBW*-triple factors

Complementary inner ideals in JBW⁎-triples

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