Currently I am finishing the reference called "C* Algebra By Example" written by Kenneth Davidson and looking for another reference related to Group C* Algebra. I tried to read the "C* Algebras and Operator Theory" written by Gerard Murray but found out that this book only focuses on representations. I used the lecture notes written by Lan Putnam before, which cover quite a number of topics in C* Algebra but does not contain that many details.

Could anyone share some of the reference you used or liked if you have ever studied Group C* Algebra? Ideally your reference will cover C* Algebra on a locally compact group or free group (e.g. $\mathbb{F}_{2}$).


I guess the answer to this question will be very different depending on whether you want material covering topics on locally compact group (i.e.: CCR, type I groups, etc) or material covering discrete groups, whose group algebras are more complex.

The flavor of the continuous case is also much more classical, while for discrete groups there are many topics that are currently (or recently) active (approximation properties, Kazhdan property for $C^\ast$-algebras, $C^\ast$-simplicity, etc)

You have books like:

  • Dixmier, Jacques, $C^*$-algebras. North-Holland Mathematical Library. Vol. 15. Amsterdam (1977). ZBL0372.46058. Dixmier book has a few chapters dedicated to group algebras.

  • Brown, Nathanial P.; Ozawa, Narutaka, $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics (2008). ZBL1160.46001. Chapters !2 and 15.

  • $\begingroup$ Before I start the learning, I would like to know some basics of those topics. I am more interested in C* Algebra on locally compact group and discrete group (maybe free group is the wrong name ...). Also, could you explain what "continuous case" here mean? And what are the topics your references focus on? $\endgroup$ – Sanae Kochiya May 23 at 15:26
  • $\begingroup$ By continuous case I meant locally compact. $\endgroup$ – Adrián González-Pérez May 24 at 11:58

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