Abstract
In this chapter we adapt a technique, borrowed from the theory of II1 factors, of juxtaposing the GNS Hilbert space structure associated with a tracial state and the C∗-algebra structure to study reduced group C∗-algebras. An emphasis is given to the C∗-algebras associated to free products of groups. We give basic norm estimates for the elements of a group algebra and present basics of Powers groups and criteria for simplicity of reduced group C∗-algebras. The chapter concludes with a study of normalizers of diffuse masas.
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Notes
- 1.
It is not difficult to check that all compact operators belong to C∗(s): start with 1 − ss∗.
- 2.
- 3.
As a matter of fact, to the best of my knowledge nobody else ever bothers to emphasize the distinction. This practice is mostly harmless because this terminology is typically used in the context of monotracial algebras, especially II1 factors.
- 4.
The projection p is not required to belong to A.
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Farah, I. (2019). Tracial States and Representations of C∗-algebras. In: Combinatorial Set Theory of C*-algebras. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-27093-3_4
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DOI: https://doi.org/10.1007/978-3-030-27093-3_4
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