1. Estimations of the numerical index of a JB$^*$-triple
- Author
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Cabezas, David and Peralta, Antonio M.
- Subjects
Mathematics - Operator Algebras ,Mathematics - Functional Analysis - Abstract
We prove that every commutative JB$^*$-triple has numerical index one. We also revisit the notion of commutativity in JB$^*$-triples to show that a JBW$^*$-triple $M$ has numerical index one precisely when it is commutative, while $e^{-1}\leq n(M) \leq 2^{-1}$ otherwise. Consequently, a JB$^*$-triple $E$ is commutative if and only if $n(E^*) =1$ (equivalently, $n(E^{**}) =1$). In the general setting we prove that the numerical index of each JB$^*$-triple $E$ admitting a non-commutative element also satisfies $e^{-1}\leq n(M) \leq 2^{-1}$, and the same holds when the bidual of $E$ contains a Cartan factor of rank $\geq 2$ in its atomic part.
- Published
- 2023