Atlas: Universal enveloping $C^*$-algebras for deformed canonical commutation relations: representations and K-theory. by Daniil Proskurin

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Functional Analysis VII
September 17-26, 2001
Department of Mathematics, University of Zagreb, Croatia
Dubrovnik, Croatia

Organizers
Hrvoje Kraljevic, Zagreb, Croatia; Davor Butkovic, Zagreb, Croatia; Murali Rao, Gainesville, Florida, USA

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Universal enveloping C^*-algebras for deformed canonical commutation relations: representations and K-theory.
by
Daniil Proskurin
Kiev Taras Shevchenko University, cybernetics department, Volodymyrska, 64, 01033, Kiev, Ukraine
Coauthors: Yurii Samoilenko (Institute of mathematics, Ukrainian Acad. Sci., Kiev, Ukraine)

We study the universal enveloping C^*-algebras for q_ij-CCR defined by M. Bozejko and R. Speiher, and for twisted CCR constructed by W. Pusz and S.L. Woronowicz. Namely we consider the *-algebras defined by the following relations

a_i^*a_j=1+q_ija_ja_i^*, i, j=1, ..., d,

q

ij
=q_ji, | q_ij | = 1, -1 < q_ii < 1i ≠ j

and

a_i^*a_i=1+m² a_ia_i^*-(1-m²)
å
j < i
a_ia_i^*, a_i^*a_j=m a_ja_i^*, i ≠ j.

We show that the universal enveloping C^*-algebra for the first class of relations is isomorphic to

C^*(s_i, | s_i^*s_i=1, s_i^*s_j=q_ijs_is_j^*, i ≠ j)

and for the second:

C^*(S_i | s_i^*s_i=1-
å
j < i
s_js_j^*, s_i^*s_j=0, s_js_i=0, i ≠ j )

. For both C^*-algebras it is proved that the Fock representation is faithful. Using concrete realizations of these C^*-algebras as operators on the Fock space we desribe their K-groups.

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Date received: March 12, 2001