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Graph product Khintchine inequalities and Hecke \(C^\ast \)-algebras: Haagerup inequalities, (non)simplicity, nuclearity and exactness. (English) Zbl 1451.05199

Summary: Graph products of groups were introduced by E. R. Green [Graph products. University of Leeds (PhD Thesis) (1990)]. M. Caspers and P. Fima [J. Noncommut. Geom. 11, No. 1, 367–411 (2017; Zbl 1373.46055)] have an operator algebraic counterpart introduced and explored. In this paper we prove Khintchine type inequalities for general \(C^\ast \)-algebraic graph products which generalize results by É. Ricard and Q. Xu [J. Reine Angew. Math. 599, 27–59 (2006; Zbl 1170.46052)] on free products of \(C^\ast \)-algebras. We apply these inequalities in the context of (right-angled) Hecke \(C^\ast \)-algebras, which are deformations of the group algebra of Coxeter groups (see [M. W. Davis, The geometry and topology of Coxeter groups. Princeton, NJ: Princeton University Press (2008; Zbl 1142.20020)]). For these we deduce a Haagerup inequality which generalizes results from [U. Haagerup, Invent. Math. 50, 279–293 (1979; Zbl 0408.46046)]. We further use this to study the simplicity and trace uniqueness of (right-angled) Hecke \(C^\ast \)-algebras. Lastly we characterize exactness and nuclearity of general Hecke \(C^\ast \)-algebras.

MSC:

05C76 Graph operations (line graphs, products, etc.)
46L05 General theory of \(C^*\)-algebras
46L10 General theory of von Neumann algebras
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20C08 Hecke algebras and their representations
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