Abstract for the talk on 18.04.2018 (09:00 h)Seminar on Nonlinear Algebra
Piotr M. Hajac (IMPAN)
An equivariant pullback structure of trimmable graph C*-algebras
We introduce a class of graphs called trimmable. Then we show that the Leavitt path algebra of a trimmable graph is graded-isomorphic to a pullback algebra of simpler Leavitt path algebras and their tensor products. Next, specializing the ground ﬁeld to the ﬁeld of complex numbers and completing Leavitt path algebras to graph C*-algbras, we prove that the graph C*-algebra of a trimmable graph is U(1)-equivariantly isomorphic with an appropriate pullback C*-algebra. As a main application, we consider a trimmable graph yielding the C*-algebra C(Sˆ2n+1˙q) of the Vaksman-Soibelman quantum sphere, and use the resulting pullback structure of its gauge invariant subalgebra C(CPˆn˙q) deﬁning the quantum complex projective space to show that the generators of the even K-group of C(CPˆn˙q) are given by a Milnor connecting homomorphism applied to the (unique up to sign) generator of the odd K-group of C(Sˆ2n-1˙q) and by the generators of the even K-group of C(CPˆn-1˙q). Based on joint works with Francesco D’Andrea, Atabey Kaygun and Mariusz Tobolski.