Standard dilations of $q$-commuting tuples

Santanu Dey¹

¹ Institut für Mathematik und Informatik Ernst-Moritz-Arndt-Universität Friedrich-Ludwig-Jahn-Str. 15a 17487 Greifswald, Germany

Colloquium Mathematicum, Tome 107 (2007) no. 1, pp. 141-165.

Résumé

We study dilations of $q$-commuting tuples. Bhat, Bhattacharyya and Dey gave the correspondence between the two standard dilations of commuting tuples and here these results are extended to $q$-commuting tuples. We are able to do this when the $q$-coefficients $q_{ij}$ are of modulus one. We introduce a “maximal $q$-commuting subspace” of an $n$-tuple of operators and a “standard $q$-commuting dilation”. Our main result is that the maximal $q$-commuting subspace of the standard noncommuting dilation of a $q$-commuting tuple is the standard $q$-commuting dilation. We also introduce the $q$-commuting Fock space as the maximal $q$-commuting subspace of the full Fock space and give a formula for a projection operator onto this space. This formula helps us in working with the completely positive maps arising in our study. We prove the first version of the Main Theorem (Theorem 21) of the paper for normal tuples by applying some tricky norm estimates and then use it to prove the general version of this theorem. We also study the distribution of a standard tuple associated with the $q$-commuting Fock space and related operator spaces.

Détail

DOI : 10.4064/cm107-1-12

Keywords: study dilations q commuting tuples bhat bhattacharyya dey gave correspondence between standard dilations commuting tuples here these results extended q commuting tuples able q coefficients modulus introduce maximal q commuting subspace n tuple operators standard q commuting dilation main result maximal q commuting subspace standard noncommuting dilation q commuting tuple standard q commuting dilation introduce q commuting fock space maximal q commuting subspace full fock space formula projection operator space formula helps working completely positive maps arising study prove first version main theorem theorem nbsp paper normal tuples applying tricky norm estimates prove general version theorem study distribution standard tuple associated q commuting fock space related operator spaces