One class of $C^*$-algebras generated by a~family of partial isometries and multiplicators

A. Yu. Kuznetsova ; E. V. Patrin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2012), pp. 44-55.

Résumé

We consider a $C^*$-subalgebra of the algebra of all bounded operators on the Hilbert space of square-summable functions defined on some countable set. This algebra is generated by a family of partial isometries and the multiplier algebra isomorphic to the algebra of all bounded functions defined on the mentioned set. The operators of partial isometries satisfy relations defined by a prescribed map on the set. We show that the considered algebra is $\mathbb Z$-graduated. After that we construct the conditional expectation from the latter onto the subalgebra responding to zero. Using this conditional expectation, we prove that the algebra under consideration is nuclear.

Détail

Keywords: partial isometry, nuclear $C^*$-algebra, conditional expectation, completely positive map.

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