×

Invariants in noncommutative dynamics. (English) Zbl 1433.46045

Summary: When a compact quantum group \(H\) coacts freely on unital \(C^\ast\)-algebras \(A\) and \(B\), the existence of equivariant maps \(A \rightarrow B\) may often be ruled out due to the incompatibility of some invariant. We examine the limitations of using invariants, both concretely and abstractly, to resolve the noncommutative Borsuk-Ulam conjectures of Baum-Dąbrowski-Hajac. Among our results, we find that for certain finite-dimensional \(H\), there can be no well-behaved invariant which solves the Type 1 conjecture for all free coactions of \(H\). This claim is in stark contrast to the case when \(H\) is finite-dimensional and abelian. In the same vein, it is possible for all iterated joins of \(H\) to be cleft as comodules over the Hopf algebra associated to \(H\). Finally, two commonly used invariants, the local-triviality dimension and the spectral count, may both change in a \(\theta\)-deformation procedure.

MSC:

46L55 Noncommutative dynamical systems
46L65 Quantizations, deformations for selfadjoint operator algebras
46L67 Quantum groups (operator algebraic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baum, Paul; Connes, Alain, Chern character for discrete groups, (A Fête of Topology (1988), Academic Press: Academic Press Boston, MA), 163-232 · Zbl 0656.55005
[2] Baum, Paul F.; Dąbrowski, Ludwik; Hajac, Piotr M., Noncommutative Borsuk-Ulam-type conjectures, Banach Center Publ., 106, 9-18 (2015) · Zbl 1343.46064
[3] Baum, Paul F.; De Commer, Kenny; Hajac, Piotr M., Free actions of compact quantum groups on unital \(C^\ast \)-algebras, Doc. Math., 22, 825-849 (2017) · Zbl 1386.46055
[4] Béguin, Cédric; Bettaieb, Hela; Valette, Alain, \(K\)-theory for \(C^\ast \)-algebras of one-relator groups, K-Theory, 16, 3, 277-298 (1999) · Zbl 0932.46063
[5] Boardman, J. Michael, Stable operations in generalized cohomology, (Handbook of Algebraic Topology (1995), North-Holland: North-Holland Amsterdam), 585-686 · Zbl 0861.55008
[6] Chirvasitu, Alexandru; Dąbrowski, Ludwik; Tobolski, Mariusz, The weak Hilbert-Smith conjecture from a Borsuk-Ulam-type conjecture · Zbl 1511.57035
[7] Chirvasitu, Alexandru; Passer, Benjamin, Compact group actions on topological and noncommutative joins, Proc. Amer. Math. Soc., 146, 8, 3217-3232 (2018) · Zbl 1452.46056
[8] Dąbrowski, Ludwik, Towards a noncommutative Brouwer fixed-point theorem, J. Geom. Phys., 105, 60-65 (2016) · Zbl 1380.58006
[9] Dąbrowski, Ludwik; Hadfield, Tom; Hajac, Piotr M., Equivariant join and fusion of noncommutative algebras, SIGMA Symmetry Integrability Geom. Methods Appl., 11, Article 082 pp. (2015) · Zbl 1343.46065
[10] Dąbrowski, Ludwik; Hajac, Piotr M.; Neshveyev, Sergey, Noncommutative Borsuk-Ulam-type conjectures revisited (2019), J. Noncommut. Geom., in press · Zbl 1481.46077
[11] Dold, Albrecht, Simple proofs of some Borsuk-Ulam results, (Proceedings of the Northwestern Homotopy Theory Conference. Proceedings of the Northwestern Homotopy Theory Conference, Evanston, Ill., 1982. Proceedings of the Northwestern Homotopy Theory Conference. Proceedings of the Northwestern Homotopy Theory Conference, Evanston, Ill., 1982, Contemp. Math., vol. 19 (1983), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 65-69 · Zbl 0521.55002
[12] Dykema, Ken; Haagerup, Uffe; Rørdam, Mikael, The stable rank of some free product \(C^\ast \)-algebras, Duke Math. J., 90, 1, 95-121 (1997) · Zbl 0905.46036
[13] Dykema, Ken; Haagerup, Uffe; Rørdam, Mikael, Correction to: “The stable rank of some free product <mml:math xmlns:mml=”http://www.w3.org/1998/Math/MathML“ altimg=”si1.gif“>C⁎-algebras”, Duke Math. J., 94, 1, 213 (1998) · Zbl 0943.46034
[14] Ellwood, David Alexandre, A new characterisation of principal actions, J. Funct. Anal., 173, 1, 49-60 (2000) · Zbl 0960.46049
[15] Gardella, Eusebio; Hajac, Piotr M.; Tobolski, Mariusz; Wu, Jianchao, The local-triviality dimension of actions of compact quantum groups
[16] Hoffmann, Burkhard, A compact contractible topological group is trivial, Arch. Math. (Basel), 32, 6, 585-587 (1979) · Zbl 0398.22006
[17] Jakob, Martin, A bordism-type description of homology, Manuscripta Math., 96, 1, 67-80 (1998) · Zbl 0897.55004
[18] Matoušek, Jiří, Using the Borsuk-Ulam theorem, (Lectures on Topological Methods in Combinatorics and Geometry. Lectures on Topological Methods in Combinatorics and Geometry, Universitext (2003), Springer-Verlag: Springer-Verlag Berlin), written in Cooperation with Anders Björner and Günter M. Ziegler · Zbl 1016.05001
[19] Matsumoto, Kengo, Noncommutative three-dimensional spheres, Jpn. J. Math. (N. S.), 17, 2, 333-356 (1991) · Zbl 0752.46041
[20] Montgomery, Susan, Hopf algebras and their actions on rings, (Published for the Conference Board of the Mathematical Sciences, Washington, DC. Published for the Conference Board of the Mathematical Sciences, Washington, DC, CBMS Regional Conference Series in Mathematics, vol. 82 (1993), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0793.16029
[21] Natsume, T.; Olsen, C. L., Toeplitz operators on noncommutative spheres and an index theorem, Indiana Univ. Math. J., 46, 4, 1055-1112 (1997) · Zbl 0906.46054
[22] Passer, Benjamin W., A noncommutative Borsuk-Ulam theorem for Natsume-Olsen spheres, J. Operator Theory, 75, 2, 337-366 (2016) · Zbl 1399.46105
[23] Passer, Benjamin, Free actions on C*-algebra suspensions and joins by finite cyclic groups, Indiana Univ. Math. J., 67, 1, 187-203 (2018) · Zbl 1411.46050
[24] Phillips, N. Christopher, Freeness of actions of finite groups on \(C^\ast \)-algebras, (Operator Structures and Dynamical Systems. Operator Structures and Dynamical Systems, Contemp. Math., vol. 503 (2009), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 217-257 · Zbl 1194.46097
[25] Robbie, Desmond; Svetlichny, Sergey, An answer to A. D. Wallace’s question about countably compact cancellative semigroups, Proc. Amer. Math. Soc., 124, 1, 325-330 (1996) · Zbl 0843.22001
[26] Taghavi, Ali, A Banach algebraic approach to the Borsuk-Ulam theorem, Abstr. Appl. Anal., Article 729745 pp. (2012) · Zbl 1247.46039
[27] Valette, Alain, Introduction to the Baum-Connes Conjecture, Lectures in Mathematics ETH Zürich (2002), Birkhäuser Verlag: Birkhäuser Verlag Basel, from notes taken by Indira Chatterji, with an appendix by Guido Mislin · Zbl 1136.58013
[28] Volovikov, A. Yu., Coincidence points of mappings of \(Z_p^n\)-spaces, Izv. Ross. Akad. Nauk Ser. Mat., 69, 5, 53-106 (2005) · Zbl 1105.55001
[29] Yamashita, Makoto, Equivariant comparison of quantum homogeneous spaces, Comm. Math. Phys., 317, 3, 593-614 (2013) · Zbl 1260.19002
[30] Yang, Xiao-Song, Construction of smooth sphere maps with given degree and a generalization of Morse index formula for smooth vector fields, Qual. Theory Dyn. Syst., 14, 1, 139-147 (2015) · Zbl 1318.58002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.