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Q-MEASURES ON THE DUAL UNIT BALL OF A JB-TRIPLE

Edwards, C. Martin;Oliveira, Lina

  • Received : 2018.02.28
  • Accepted : 2018.08.02
  • Published : 2019.01.01

Abstract

Let A be a $JB^*$-triple with Banach dual space $A^*$ and bi-dual the $JBW^*$-triple $A^{**}$. Elements x of $A^*$ of norm one may be regarded as normalised 'Q-measures' defined on the complete ortho-lattice ${\tilde{\mathcal{U}}}(A^{**})$ of tripotents in $A^{**}$. A Q-measure x possesses a support e(x) in ${\tilde{\mathcal{U}}}(A^{**})$ and a compact support $e_c(x)$ in the complete atomic lattice ${\tilde{\mathcal{U}}}_c(A)$ of elements of ${\tilde{\mathcal{U}}}(A^{**})$ compact relative to A. Necessary and sufficient conditions for an element v of ${\tilde{\mathcal{U}}}_c(A)$ to be a compact support tripotent $e_c(x)$ are given, one of which is related to the Q-covering numbers of v by families of elements of ${\tilde{\mathcal{U}}}_c(A)$.

Keywords

$JB^*$-triple;$C^*$-algebra;Q-topology;Q-measure;compact support tripotent

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Acknowledgement

Supported by : FCT/Portugal