## Illinois Journal of Mathematics

### Remarks on HNN extensions in operator algebras

Yoshimichi Ueda

#### Abstract

It is shown that any HNN extension is precisely a compression by a projection of a certain amalgamated free product in the framework of operator algebras. As its applications several questions for von Neumann algebras or C*-algebras arising as HNN extensions are considered.

#### Article information

Source
Illinois J. Math., Volume 52, Number 3 (2008), 705-725.

Dates
First available in Project Euclid: 28 September 2009

https://projecteuclid.org/euclid.ijm/1254143997

Digital Object Identifier
doi:10.1215/ijm/1254143997

Mathematical Reviews number (MathSciNet)
MR2546003

Zentralblatt MATH identifier
1183.46057

#### Citation

Ueda, Yoshimichi. Remarks on HNN extensions in operator algebras. Illinois J. Math. 52 (2008), no. 3, 705--725. doi:10.1215/ijm/1254143997. https://projecteuclid.org/euclid.ijm/1254143997

#### References

• J. Anderson and W. L. Paschke, The $K$-theory of the reduced $C^*$-algebra of an HNN-group, J. Operator Theory 16 (1986), 165–187.
• D. Avitzour, Free products of $C^*$-algebras, Trans. Amer. Math. Soc. 271 (1982), 423–465.
• E. F. Blanchard and K. J. Dykema, Embeddings of reduced free products of operator algebras, Pacific J. Math. 199 (2001), 1–19.
• N. Brown, K. Dykema and K. Jung, Free entropy dimension in amalgamated free products, with an appendix by W. Lück, to appear in Proc. Lond. Math. Soc., available at math.OA/0609080.
• P. de la Harpe, On simplicity of reduced $C^*$-algebras of groups, Bull. Lond. Math. Soc. 39 (2007), 1–26.
• J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, II, Trans. Amer. Math. Soc. 234 (1977), 289–324.
• D. Gaboriau, Coût des relations d'équivalence et des groupes, Invent. Math. 139 (2000), 41–98.
• K. Jung, The free entropy dimension of hyperfinite von Neumann algebras, Trans. Amer. Math. Soc. 355 (2003), 5053–5089.
• W. Lück, Survey on classifying spaces for families of subgroups, Infinite groups: Geometric, combinatorial and dynamical aspects, Progress in Mathematics, vol. 248, Birkhäuser, Basel, 2005.
• K. McClanahan, Simplicity of reduced amalgamated products of $C\sp *$-algebras, Canad. J. Math. 46 (1994), 793–807.
• F. Paulin, Propriétés asymptotiques des relations d'équivalences mesurées discrètes, Markov Process. Related Fields 5 (1999), 163–200.
• M. Pimsner and D. Voiculescu, $K$-groups of reduced crossed products by free groups, J. Operator Theory 8 (1982), 131–156.
• S. Popa, On a problem of R. V. Kadison on maximal abelian $*$-subalgebras in factors, Invent. Math. 65 (1981/82), 269–281.
• F. Rădulescu, The von Neumann algebra of the non-residually finite Baumslag group $\langle a,b | ab^3 a^{-1} = b^2 \rangle$ embeds into $R^\omega$, Hot topics in operator theory, Theta Ser. Adv. Math., vol. 9, Theta, Bucharest, 2008, pp. 173–185.
• M. Takesaki, Theory of operator algebras, II, Encyclopaedia of Mathematical Sciences, vol. 125, Operator Algebras and Non-commutative Geometry, vol. 6, Springer-Verlag, Berlin, 2003.
• K. Thomsen, On the $KK$-theory and the $E$-theory of amalgamated free products of $C\sp *$-algebras, J. Funct. Anal. 201 (2003), 30–56.
• Y. Ueda, Amalgamated free product over Cartan subalgebra, Pacific J. Math. 191 (1999), 359–392.
• Y. Ueda, Fullness, Connes' $\chi$-groups, and ultra-products of amalgamated free products over Cartan subalgebras, Trans. Amer. Math. Soc. 355 (2003), 349–371.
• Y. Ueda, HNN extensions of von Neumann algebras, J. Funct. Anal. 225 (2005), 383–426.\goodbreak
• D. Voiculescu, K. Dykema and A. Nica, Free random variables, CRM Monograph Series, vol. 1, Amer. Math. Soc., Providence, RI, 1992.