Rocky Mountain Journal of Mathematics

The embedding of $\mathcal{O}_{d_1d_2}$ into $\mathcal{O}_{d_1}\otimes\mathcal{O}_{d_2}$

Amy B. Chambers

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As a particular example of a general theorem presented in \cite{me}, there is a conditional expectation from the tensor product of Cuntz algebras, $\mathcal{O}_{d_1}\otimes\mathcal{O}_{d_2}$, onto the Cuntz algebra $\mathcal{O}_{d_1d_2}$. Motivated by this example, we examine the embedding of $\mathcal{O}_{d_1d_2}$ in $\mathcal{O}_{d_1}\otimes\mathcal{O}_{d_2}$, first by examining the index of the conditional expectation mentioned, and then by expressing $\mathcal{O}_{d_1}\otimes\mathcal{O}_{d_2}$ as a concrete Paschke crossed product by an endomorphism and then abstractly as the image of a faithful representation of the Stacey crossed product of $\mathcal{O}_{d_1d_2}$ by the same endomorphism.

Article information

Rocky Mountain J. Math., Volume 44, Number 4 (2014), 1111-1124.

First available in Project Euclid: 31 October 2014

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Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras

Graph algebra conditional exp ectation wavelet Paschke crossed product by an endomorphism Stacy crossed pro duct Cuntz algebra subalgebra


Chambers, Amy B. The embedding of $\mathcal{O}_{d_1d_2}$ into $\mathcal{O}_{d_1}\otimes\mathcal{O}_{d_2}$. Rocky Mountain J. Math. 44 (2014), no. 4, 1111--1124. doi:10.1216/RMJ-2014-44-4-1111.

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  • Sarah Boyd, Navin Keswani and Iain Raeburn, Faithful representations of crossed products by endomorphisms, Proc. Amer. Math. Soc. 118 (1993), 427–436.
  • Amy Chambers, Certain subalgebras of the tensor product of graph algebras, J. Oper. Theor. 63 (2010), 389–402.
  • Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173–185.
  • David S. Dummit and Richard M. Foote, Abstract algebra, Prentice Hall Inc., Englewood Cliffs, NJ, 1991.
  • Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.
  • Gerard J. Murphy, $C^*$-algebras and operator theory, Academic Press Inc., Boston, MA, 1990.
  • William L. Paschke, The crossed product of a $C*$-algebra by an endomorphism, Proc. Amer. Math. Soc. 80 (1980), 113–118.
  • Iain Raeburn and Astrid an Huef, preprint, June 2010.
  • M. Rørdam, Classification of nuclear, simple $C^*$-algebras,in Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encycl. Math. Sci. 126, Springer, Berlin, 2002.
  • M. Rørdam, F. Larsen and N. Laustsen, An introduction to $K$-theory for $C^*$-algebras, Lond. Math. Soc. Student Texts 49, Cambridge University Press, Cambridge, 2000.
  • Claude Schochet, Topological methods for $C^*$-algebras II, Geometry resolutions and the Künneth formula, Pac. J. Math. 98 (1982), 443–458.
  • P.J. Stacey, Crossed products of $c^*$-algebras by $*$-endomorphisms, J. Austr. Math. Soc. 54 (1993), 204–212.
  • Y. Watatani, e-mail correspondence to Judith Packer, July 2005.
  • Yasuo Watatani, Index for $C^*$-subalgebras, Mem. Amer. Math. Soc. 83 (1990), vi+117.
  • N.E. Wegge-Olsen, $K$-theory and $C^*$-algebras, Oxford Sci. Publ., The Clarendon Press, Oxford University Press, New York, 1993.