The rotation algebra. (English) Zbl 0703.22005

Anderson, Joel; Paschke, William

Houston J. Math. 15, No. 1, 1-26 (1989).

The authors consider the rotation algebra A which is the group \(C^*\)- algebra \(A:=C^*(H)\) associated to the discrete three-dimensional Heisenberg group H. They show that A is isomorphic to the algebra of continuous sections of a field of \(C^*\)-algebras. The authors present some result concerning the irreducible representations of A and show that there exists a “small” separating family of finite-dimensional representations. Next the authors obtain a variant of a theorem of N. Riedel [J. Oper. Theory 13, 143-150 (1985; Zbl 0607.46038)] concerning the density of the invertible elements in irrational rotation algebras. The proof of this theorem seems to be simpler and more natural since it avoids the introduction of diophantine approximations used by N. Riedel. The final section is devoted to a spectral calculation on irrational rotation algebras.

Reviewer: U.Grimmer

MSC:

22D25	\(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L05	General theory of \(C^*\)-algebras
46L55	Noncommutative dynamical systems

Keywords:

group \(C^ *\)-algebra; Heisenberg group; algebra of continuous sections; \(C^ *\)-algebras; irreducible representations; finite-dimensional representations; irrational rotation algebras