An equivariant pullback structure of trimmable graph C ∗ C^* C ∗ -algebras

[摘要] To unravel the structure of fundamental examples studied in noncommutative topology, we prove that the graph C∗C^*C∗-algebra C∗(E)C^*(E)C∗(E) of a trimmable graph EEE is U(1)U(1)U(1)-equivariantly isomorphic to a pullback C∗C^*C∗-algebra of a subgraph C∗C^*C∗-algebra C∗(E′′)C^*(E'')C∗(E′′) and the C∗C^*C∗-algebra of functions on a circle tensored with another subgraph C∗C^*C∗-algebra C∗(E′)C^*(E')C∗(E′). This allows us to approach the structure and K-theory of the fixed-point subalgebra C∗(E)U(1)C^*(E)^{U(1)}C∗(E)U(1) through the (typically simpler) C∗C^*C∗-algebras C∗(E′)C^*(E')C∗(E′), C∗(E′′)C^*(E'')C∗(E′′) and C∗(E′′)U(1)C^*(E'')^{U(1)}C∗(E′′)U(1). As examples of trimmable graphs, we consider one-loop extensions of the standard graphs encoding respectively the Cuntz algebra O2\mathcal{O}_2O2​ and the Toeplitz algebra T\mathcal{T}T. Then we analyze equivariant pullback structures of trimmable graphs yielding the C∗C^*C∗-algebras of the Vaksman–Soibelman quantum sphere Sq2n+1S^{2n+1}_qSq2n+1​ and the quantum lens space Lq3(l;1,l)L_q^3(l;1,l)Lq3​(l;1,l), respectively.

[学科分类] 神经科学

[关键词] Compact quantum spaces;graphC -algebras;equivariant pullbacks;K-theory;the Mayer–Vietoris sequence