Quartic Algebras

Abstract

In this paper we study the fixed point algebra of the automorphism of the rotation algebra , θ = p/q with p, q coprime positive integers, given by u → v-1, v → u. We give a general characterization of the fixed point algebra, determine its K-theory and consider the related crossed-product algebra ⋊Ƭ Z4.

Toroidal orbifolds of Z3 and Z6 symmetries of noncommut

“…(1.11) 9 We point out that the topological invariants listed in Table 2 of [3] are those of the above mentioned maps ψ θ k which differ from the maps we used in [3] by normalization constants -particularly for the maps ϕ 11 , ϕ 12 used in [3] which involved the constants e(− θ 6 ), e(− 2θ 3 ), respectively, and which should be removed (as we have in fact done so at the end of Section 9 of [3]). 10 In [3], the unbounded trace values differ by a factor of 3 since we were working with the crossed product C * -algebra A θ κ Z 3 , but since this algebra is strongly Morita equivalent to the fixed point C * -subalgebra A κ θ the unbounded trace values in Table 2 of [3] need to be multiplied by 3.…”

Fixed point subalgebras of rational higher-dimensional non-commutative tori

“…Firstly recall the following explicit description of the rotation algebra A p q , (p, q) = 1, see for example [15,4,11,12,13].…”

“…For θ rational, θ = p q , there is a complete description of the fixed point subalgebras A Zi θ for all four cases i = 2, 3, 4, 6 [3,10,11,12,13], as trivial matrix bundle C * -algebras over the 2-sphere S 2 with generic fiber M q and exceptional fibers at four points of S 2 for i = 2 and three points of S 2 for i = 3, 4, 6 (at least for q > i; for q ≤ i an analogous description is true but there are fewer fixed points and their orders could be smaller). In fact a similar description is also valid for the crossed products A θ Z i .…”

“…For the 2-dimensional non-commutative torus or rotation algebra, A θ , the situation has been considered in some depth [3,4,5,6,9,10,11,12,13,14,15,16,17]. The automorphism group of irrational rotation algebras (up to inner automorphisms) has recently been described by Elliott and Rørdam [9] as generated by the automorphisms induced by SL(2, Z) and an automorphism inducing a determinant −1 transformation on K 1 (A θ ) ∼ = Z 2 (of which there is no explicit description).…”

Projections in Rotation Algebras and Theta Functions

“…We notice that for any α ∈ R \ Q, the C * -algebra A σ α is generated only by the operators H α = H (1) α and H (2) α (see the appendix). The K-groups of A σ α were computed in the case α = p q ∈ Q, gcd(p, q) = 1 in [7], where it was shown that K 1 (A σ α ) = 0 and K 0 (A σ α ) = Z 9 if q ≥ 5. The problem of characterizing the spectrum of the self-adjoint operator H α = u + u * + v + v * (or more generally of H α,λ = u + u * + λ 2 (v + v * ), λ > 0) in A α is very important in the study of the quantum Hall effect ( [2]).…”