#### Vol. 288, No. 2, 2017

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Topological invariance of quantum quaternion spheres

### Bipul Saurabh

Vol. 288 (2017), No. 2, 435–452
##### Abstract

The ${C}^{\ast }$-algebra of continuous functions on the quantum quaternion sphere ${H}_{q}^{2n}$ can be identified with the quotient algebra $C\left({SP}_{q}\left(2n\right)∕{SP}_{q}\left(2n-2\right)\right)$. In the commutative case, i.e., for $q=1$, the topological space $SP\left(2n\right)∕SP\left(2n-2\right)$ is homeomorphic to the odd-dimensional sphere ${\mathsc{S}}^{4n-1}$. In this paper, we prove the noncommutative analogue of this result. Using homogeneous ${C}^{\ast }$-extension theory, we prove that the ${C}^{\ast }$-algebra $C\left({H}_{q}^{2n}\right)$ is isomorphic to the ${C}^{\ast }$-algebra $C\left({S}_{q}^{4n-1}\right)$. This further implies that for different values of $q$ in $\left[0,1\right)$, the ${C}^{\ast }$-algebras underlying the noncommutative spaces ${H}_{q}^{2n}$ are isomorphic.

##### Keywords
homogeneous extension, quantum double suspension, corona factorization property
##### Mathematical Subject Classification 2010
Primary: 19K33, 46L80, 58B34