Taiwanese Journal of Mathematics

THE CROSSED PRODUCT VON NEUMANN ALGEBRAS ASSOCIATED WITH $SL_2(\mathbb{R})$

Wenming Wu and Wei Yuan

Full-text: Open access

Abstract

Let $\mathcal{A}$ be the abelian von Neumann subalgebra $\{M_{f}: f \in L^{\infty}(\mathbb{H},\mu_{r})\}$ of $\mathcal{B}(L^{2}(\mathbb{H},\mu_{r}))$, where $\mathbb{H}$ is the upper half plane and the measure $d\mu_{r} = dx dy/y^{2-r}$. For any integers $r \gt 1$, the linear fractional action of $SL_{2}(\mathbb{R})$ on $\mathbb{H}$ induces a continuous action $\alpha$ of $SL_{2}(\mathbb{R})$ on $\mathcal{A}$. It is shown that the crossed product $\mathcal{R}(\mathcal{A}, \alpha)$ of $\mathcal{A}$ under the action $\alpha$ of $SL_{2}(\mathbb{R})$ is *-isomorphic to $\mathcal{B}(L^{2}(P, 2dx dy/y^{3-2r})) \overline{\otimes} \mathcal{L}_{K}$, where $SL_{2}(\mathbb{R}) = PK$ is the Iwasawa decomposition of $SL_{2}(\mathbb{R})$. Thus $\mathcal{R}(\mathcal{A}, \alpha)$ is of type $\mathrm{I}$.

Article information

Source
Taiwanese J. Math., Volume 14, Number 4 (2010), 1501-1515.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405964

Digital Object Identifier
doi:10.11650/twjm/1500405964

Mathematical Reviews number (MathSciNet)
MR2663928

Subjects
Primary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx] 46L99: None of the above, but in this section

Keywords
crossed product von Neumann algebras Iwasawa decomposition $SL_{2}(\mathbb{R})$ linear fractional action

Citation

Wu, Wenming; Yuan, Wei. THE CROSSED PRODUCT VON NEUMANN ALGEBRAS ASSOCIATED WITH $SL_2(\mathbb{R})$. Taiwanese J. Math. 14 (2010), no. 4, 1501--1515. doi:10.11650/twjm/1500405964. https://projecteuclid.org/euclid.twjm/1500405964


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References

  • V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. Math., 48 (1947), 568-640.
  • R. Kadison and J. Ringrose, Fundamentals of the theory of operator algebras, Vols. I and II, Academic Press, Orlando, 1983 and 1986.
  • S. Lang, $SL_{2}(\mathbb{R})$, Springer-Verlag, New York, 1985.
  • F. R\vaa dulescu, The $\Gamma$-equivariant form of the Berezin quantization of the upper half plane, Memoirs of A.M.S. 630, Vol. 133, Rhode Island, 1998.
  • M. Takesaki, Theory of operator algebra $\mathrm{II}$, Springer-Verlag, Berlin, 2003.
  • A. Van. Daele, Continuous crossed products and Type III von Neumann algebras, Camb. Univ. Press, Cambridge, 1978.
  • Wenming Wu, A note on the crossed product $\mathcal{R}(\mathcal{A},\alpha)$ associated with $PSL_{2}(\mathbb{R})$, Science in China, Series A: Mathematic, 51(11) (2008), 2081-2088.