Compact group actions on operator algebras and their spectra

  • Costel Peligrad


We consider a class of dynamical systems with compact non-abelian groups that include C*-, W*- and multiplier dynamical systems. We prove results that relate the algebraic properties such as simplicity or primeness of the fixed point algebras to the spectral properties of the action, including the Connes and strong Connes spectra.


Arveson, W., On groups of automorphisms of operator algebras, J. Functional Analysis 15 (1974), 217–243.

Connes, A., Une classification des facteurs de type $\rm III$, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252.

Connes, A. and Takesaki, M., The flow of weights on factors of type $\rm III$, Tôhoku Math. J. (2) 29 (1977), no. 4, 473–575.

D'Antoni, C. and Zsidó, L., Groups of linear isometries on multiplier $C^*$-algebras, Pacific J. Math. 193 (2000), no. 2, 279–306.

Dumitru, R. and Peligrad, C., Compact quantum group actions on $C^*$-algebras and invariant derivations, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3977–3984.

Gootman, E. C., Lazar, A. J., and Peligrad, C., Spectra for compact group actions, J. Operator Theory 31 (1994), no. 2, 381–399.

Hille, E. and Phillips, R. S., Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957.

Kishimoto, A., Simple crossed products of $C^\ast $-algebras by locally compact abelian groups, Yokohama Math. J. 28 (1980), no. 1-2, 69–85.

Landstad, M. B., Algebras of spherical functions associated with covariant systems over a compact group, Math. Scand. 47 (1980), no. 1, 137–149.

Olesen, D., Inner $^\ast $-automorphisms of simple $C^\ast $-algebras, Comm. Math. Phys. 44 (1975), no. 2, 175–190.

Olesen, D. and Pedersen, G. K., Applications of the Connes spectrum to $C^\ast $-dynamical systems, J. Functional Analysis 30 (1978), no. 2, 179–197.

Olesen, D., Pedersen, G. K., and Størmer, E., Compact abelian groups of automorphisms of simple $C^*$-algebras, Invent. Math. 39 (1977), no. 1, 55–64.

Pedersen, G. K., $C^\ast $-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, London-New York, 1979.

Peligrad, C., Locally compact group actions on $C^*$-algebras and compact subgroups, J. Funct. Anal. 76 (1988), no. 1, 126–139.

Peligrad, C., A solution of the maximality problem for one-parameter dynamical systems, Adv. Math. 329 (2018), 742–780.

Zsidó, L., Spectral and ergodic properties of the analytic generators, J. Approximation Theory 20 (1977), no. 1, 77–138.
How to Cite
Peligrad, C. (2020). Compact group actions on operator algebras and their spectra. MATHEMATICA SCANDINAVICA, 126(2), 276-292.