Elliptic Algebras

Farsi, Carla; Watling, Neil

doi:10.1006/jfan.1993.1136

Cited by 7 publications

(11 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…10 Observe that the range in (1.8) is independent of the "Planck" parameter θ , whereas the canonical bounded trace τι has range τιK 0 (A κ θ ) = Z + Zθ which depends on θ .…”

Section: ρ(E(t)) = E(t) κ(E(t)) = E(t);mentioning

confidence: 98%

“…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. For the Hexic case one similarly multiplies the unbounded trace values in Table 1 (Here, of course, ρ 2 = κ is the Cubic and ρ 3 = φ is the Flip.)…”

Section: Remark 13mentioning

confidence: 99%

See 1 more Smart Citation

Toroidal orbifolds of Z3 and Z6 symmetries of noncommut

Walters

2015

Nuclear Physics B

View full text Add to dashboard Cite

The Hexic transform ρ of the noncommutative 2-torus A θ is the canonical order 6 automorphism defined by ρ(U ) = V , ρ(V ) = e −πiθ U −1 V , where U , V are the canonical unitary generators obeying the unitary Heisenberg commutation relation VU = e 2πiθ UV. The Cubic transform is κ = ρ 2 . These are canonical analogues of the noncommutative Fourier transform, and their associated fixed point C * -algebras A ρ θ , A κ θ are noncommutative Z 6 , Z 3 toroidal orbifolds, respectively. For a large class of irrationals θ and rational approximations p/q of θ , a projection e of trace q 2 θ − pq is constructed in A θ that is invariant under the Hexic transform. Further, this projection is shown to be a matrix projection in the sense that it is approximately central, the cut down algebra eA θ e contains a Hexic invariant q × q matrix algebra M whose unit is e and such that the cut downs eUe, eVe are approximately inside M. It is also shown that these invariant matrix projections are covariant in that they arise from a continuous section E(t) of C ∞ -projections of the continuous field {A t } 0

show abstract

“…10 Observe that the range in (1.8) is independent of the "Planck" parameter θ , whereas the canonical bounded trace τι has range τιK 0 (A κ θ ) = Z + Zθ which depends on θ .…”

Section: ρ(E(t)) = E(t) κ(E(t)) = E(t);mentioning

confidence: 98%

Section: Remark 13mentioning

confidence: 99%

Toroidal orbifolds of Z3 and Z6 symmetries of noncommut

Walters

2015

Nuclear Physics B

View full text Add to dashboard Cite

show abstract

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

Section: An Alternative Description For the Algebra A θmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

“…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 .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation