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On the K-theory of the \(C^*\)-algebra generated by a projective representation of a torsion-free discrete abelian group. (English) Zbl 0542.46030

Operator algebras and group representations, Proc. int. Conf., Neptun/Rom. 1980, Vol. I, Monogr. Stud. Math. 17, 157-184 (1984).
[For the entire collection see Zbl 0515.00017.]
The range of the trace on \(K_ 0\) of the irrotational rotation \(C^*\)- algebra \(A_{\theta}\) (generated by unitaries \(u_ 1\) and \(u_ 2\) with commutation relation \(u_ 2u_ 1=e^{2\pi i\theta}u_ 1u_ 2)\) was computed by the combined work of M. A. Rieffel [Pac. J. Math. 93, 415-429 (1981; Zbl 0499.46039)] and M. Pimser and D. Voiculescu [J. Oper. Theory 4, 201-210 (1980; Zbl 0525.46031)], with preliminary work by Powers (unpublished) who showed that projections exist. The answer is \({\mathbb{Z}}+{\mathbb{Z}}\theta.\)
In the present paper this invariant is computed for any \(C^*\)-algebra generated by a subsets of unitaries each two of which have such a commutation relation, i.e. by a family of unitaries \((u_{\alpha})_{\alpha \in I}\) such that for each pair \(\alpha\),\(\beta \in I\) there exists \(\theta_{\alpha \beta}\in {\mathbb{R}}\) with \(u_{\beta}u_{\alpha}=e^{2\pi i\theta_{\alpha \beta}}u_{\alpha}u_{\beta}\). The subgroup of \({\mathbb{R}}\) generated by the values of a tracial state of such a \(C^*\)-algebra on projections, and on matrix projections, is the same for any tracial state, and is the subgroup of \({\mathbb{R}}\) generated by the homogeneous polynomials \[ 1,\theta_{\alpha_ 1\alpha_ 2},\quad \theta_{\alpha_ 1\alpha_ 2}\theta_{\alpha_ 3\alpha_ 4}+\theta_{\alpha_ 1\alpha_ 3}\theta_{\alpha_ 4\alpha_ 2}+\theta_{\alpha_ 1\alpha_ 4}\theta_{\alpha_ 2\alpha_ 3},..., \] where each of the polynomials of degree n is the sum of (2n-1) (2n-3)...1 monomials. If the values of \(\theta\) for different pairs of indices are chosen so that \(\theta_{\beta \alpha}=-\theta_{\alpha \beta}\), then the homogeneous polynomials of degree n arising in this sequence may be described as the components of the antisymmetric tensor \(1/n!\theta \wedge...\wedge \theta\) (n factors). The method of proof is to regard a given \(C^*\)- algebra generated by unitaries with commutation relations of this kind as a quotient of a universal \(C^*\)-algebra generated by unitaries with these relations, and then to express this universal \(C^*\)-algebra as an iteration of crossed products by \({\mathbb{Z}}\), to each of which the six term K-theory exact sequence derived by M. Pimsner and D. Voiculescu in [J. Oper. Theory 4, 93-118 (1980; Zbl 0474.46059)] can be applied. At the same time the value of \(\theta\) for each pair \(\alpha\),\(\beta \in I\) is allowed to vary as a parameter, and the irrational values are dealt with as limits of rational values, in the spirit of the original argument of Powers.
In the case that the numbeer of unitary generators is just two, the work of Rieffel referred to above established more: all real numbers in this subgroup between 0 and 1 are actually values of the trace on projections. It is not difficult to extend this result to the case of three unitary generators. Recently M. A. Rieffel has given a new construction of projections which may allow this result to be extended [Proceedings of the Japan-U.S. Seminar ”Geometric Methods in Operator Algebras”, Kyoto University, 1983].

MSC:

46L05 General theory of \(C^*\)-algebras
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
46L55 Noncommutative dynamical systems