AMS Subject Classification No.: 19K33, 46L80, 58B34.
Keywords. \(C^*\)-extension, homogeneous extension, corona factorization property, \(m\)-torsioned quantum double suspension.

1 Introduction↩︎

Hong and Szymanski introduced the notion of quantum double suspension (QDS) for unital \(C^*\)-algebras in [1]. This construction allows one to pass from classical compact spaces to noncommutative ones in a systematic way. When a noncommutative space is obtained by iteratively applying the QDS to a compact space \(X\), many of its topological and geometric properties can often be understood in terms of those of \(X\). For instance, it simplifies the computation of \(K\)-groups and produces examples of noncommutative geometries from the geometries of the classical space (see [2]). Similar to QDS, for any unital \(C^*\)-algebra \(A\), the \(m\)-torsioned quantum double suspension, denoted by \(\Sigma^2_m A\), is defined in [3]. For \(m=1\), this construction reduces to the usual QDS. As in the case of QDS, it is desirable to realize a given \(C^*\)-algebra as an iterated \(m\)-torsioned quantum double suspension of \(C(X)\) for some compact space \(X\). In [3], it was shown that the \(C^*\)-algebra \(C(SO_q(3))\) is isomorphic to \(\Sigma^2_2 C(\mathbb{T})\). The purpose of the present article is to generalize this result and show that the \(C^*\)-algebra associated with the quantum homogeneous space \(SO_q(2n+1)/SO_q(2n-1)\) can be obtained from \(C(\mathbb{T})\) by successively applying the QDS and the \(2\)-torsioned QDS in a suitable order. More precisely, we establish an isomorphism between \(C\!\left(SO_q(2n+1)/SO_q(2n-1)\right)\) and \(\Sigma^{2(n-1)} \, \Sigma^2_2 \, \Sigma^{2(n-1)} C(\mathbb{T})\). This description immediately implies that the topological type of the underlying \(C^*\)-algebra is independent of the deformation parameter \(q\).

We now briefly outline the main idea. For a nuclear \(C^*\)-algebra \(A\) and a finite-dimensional compact metric space \(Y\) (that is, a closed subset of \(\mathcal{S}^n\) for some \(n \in \mathbb{N}\)), Pimsner, Popa, and Voiculescu [4] introduced a group \(\operatorname{Ext}_{\mathrm{PPV}}(Y, A)\) consisting of strongly unitary equivalence classes of homogeneous extensions of \(A\) by \(C(Y) \otimes \mathcal{K}\). An important feature of this group, which distinguishes it from the usual \(Ext\) group, is that any two elements representing the same class must have isomorphic middle \(C^*\)-algebras. This property plays a crucial role in the present work. We first establish an isomorphism between \(\operatorname{Ext}_{\mathrm{PPV}}(Y, A)\) and \(\operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma^2_m A)\), under mild assumptions on \(A\) and \(Y\). Taking \(Y = \mathbb{T}\), we then show that \(K_0\) serves as a complete invariant for the class of \(C^*\)-algebras arising as middle algebras of homogeneous extensions of \(\Sigma^2_2 \Sigma^{2(n-1)} C(\mathbb{T})\) by \(C(\mathbb{T}) \otimes \mathcal{K}\). The result follows by comparing the \(K\)-groups, and then extending it using the main result of [5].

Our article is organized as follows. In Section 2, we recall the construction of \(m\)-torsioned quantum double suspensions, and the quotient space \(SO_q(2n+1)/SO_q(2n-1)\), and and some related results. In Section 3, we establish the isomorphism between \(\operatorname{Ext}_{\mathrm{PPV}}(\mathbb{T}, A)\) and \(\operatorname{Ext}_{\mathrm{PPV}}(\mathbb{T}, \Sigma^2_mA)\). Using this, we prove that \(C\!\left(SO_q(2n+1)/SO_q(2n-1)\right)\) is isomorphic to \(\Sigma^{2(n-1)}\Sigma^2_2 \Sigma^{2(n-1)} C(\mathbb{T})\).

We now fix some notation. Throughout, the symbol \(q\) denotes a real number in the interval \((0,1)\). The standard orthonormal bases of the Hilbert spaces \(\ell^{2}(\mathbb{N})\) and \(\ell^{2}(\mathbb{Z})\) are denoted by \(\{e_n : n \in \mathbb{N}\}\) and \(\{e_n : n \in \mathbb{Z}\}\), respectively. The left shift operator on both \(\ell^{2}(\mathbb{N})\) and \(\ell^{2}(\mathbb{Z})\) will be denoted by the same symbol \(S\). For \(m<0\), the expression \((S^{*})^{m}\) denotes the operator \(S^{-m}\). Let \(p_{ji}\) denote the rank-one operator sending \(e_i\) to \(e_j\). We write \(p_{ii}\) simply as \(p_i\), and \(p_{00}\) as \(p\). The symbol \(p_{<m}\) denotes the projection \(\sum_{i=0}^{m-1} p_i\). We write \(\mathcal{L}(\mathcal{H})\) and \(\mathcal{K}(\mathcal{H})\) for the sets of all bounded linear operators and all compact operators on a Hilbert space \(\mathcal{H}\), respectively, and denote by \(\mathcal{K}\) the \(C^{*}\)-algebra of compact operators. For a \(C^{*}\)-algebra \(A\), we use \(M(A)\) and \(Q(A)\) to denote its multiplier algebra and corona algebra, respectively. The map \(\pi\) denotes the canonical homomorphism from \(M(A)\) onto \(Q(A)\), and for \(a \in M(A)\), the symbol \([a]\) denotes its image under \(\pi\). For a group \(G\), let \(G^{\mathrm{Tor}}\) denote the torsion subgroup of \(G\).

2 Preliminaries↩︎

We begin by recalling the definition and certain known results concerning the \(m\)-torsioned quantum double suspension and the quotient space \(SO_q(2n+1)/SO_q(2n-1)\), which will be used in the later sections.

2.1 \(m\)-torsioned Quantum double suspension↩︎

The following proposition establishes the universal property of \(m\)-torsioned quantum double suspension.

: Given a representation \(\rho\) of \(A\) acting on the Hilbert space \(\mathcal{H}\), one can define a representation \(\Sigma^2_m \rho\) of \(\Sigma^2_m A\) actiong on \(\mathcal{H}\otimes \ell^2(\mathbb{N})\) as follows (see [3]). \[\Sigma^2 \rho\,(1\otimes S^*)=1\otimes S^* \, \,{ and } \, \, \Sigma_m^2 \rho\,(a\otimes p)=\rho(a)\otimes p,for alla \in A.\] It is not difficult to verify that \(\ker \Sigma^2_m \rho=\ker \rho \otimes \mathcal{K}\). Using this and Theorem \(5.7\) in [3], the claim follows. \(\Box\)

: Since \(\phi(a \otimes k)\neq 0\) for any \(a \in A\) and \(k \in \mathcal{K}\), it follows from Proposition 3 that the only possibility is \(\ker \phi=0\). \(\Box\)

2.2 The quotient space \(SO_q(2n+1)/SO_q(2n-1)\)↩︎

Let \[B_k^{2n+1}(q)= \begin{cases} C(\mathbb{T}) &ifk=1, \cr \phi_{\omega_k}(C(SO_q(2n+1)/SO_q(2n-1))) &if1<k \leq 2n. \cr \end{cases}\] We now recall some results from [3], which are needed to prove our main claim.

2.3 \(C^*\)-algebra extensions↩︎

Let \(A\), and \(B\) be unital separable nuclear \(C^*\)-algebras. Two elements \(a,b \in Q(B \otimes \mathcal{K})\) are said to be strongly unitarily equivalent if there exists a unitary \(U \in M(B \otimes \mathcal{K})\) such that \[[U]\, a \, [U^*] = b,\] and we write \(a \sim_{su} b\). Two \(C^*\)-extensions \(\alpha, \beta : A \to Q(B \otimes \mathcal{K})\) are said to be strongly unitarily equivalent, denoted by \(\alpha\sim_{su}\beta\), if there exists a unitary \(U \in M(B \otimes \mathcal{K})\) such that \[, [U]\, \alpha(a) \, [U^*] = \beta(a) \quad \text{for all } a \in A.\] An element \(a\) in a \(C^*\)-algebra \(A\) is called norm-full if it is not contained in any proper closed ideal of \(B\). An extension \[\tau : A \to Q(B \otimes \mathcal{K})\] is said to be norm-full if for every nonzero element \(a \in A\), the element \(\tau(a)\) is norm-full in \(Q(B \otimes \mathcal{K})\).

It is easy to see that if a \(C^*\)-algebra \(B\) has the corona factorization property, then any norm-full projection in \(Q(B)\) is Murray-von Neumann equivalent to the unit of \(Q(B)\). Furthermore, one can show that for a finite-dimensional compact metric space \(Y\), the algebra \(C(Y) \otimes \mathcal{K}\) has the corona factorization property (see [4]).

Suppose that \(Y\) is a finite-dimensional compact metric space. In other words, \(Y\) is homeomorphic to a closed subset of the Euclidean sphere \(\mathcal{S}^n\) for some \(n \in \mathbb{N}\). Let \[M(Y) := M(C(Y) \otimes \mathcal{K}), \quad Q(Y) := M(C(Y) \otimes \mathcal{K}) / (C(Y) \otimes \mathcal{K}),\] and let \(Q := \mathcal{L}(\mathcal{H}) / \mathcal{K}(\mathcal{H})\) denote the Calkin algebra. It is easy to see that \(M(Y)\) can be identified with the set of all \(*\)-strongly continuous functions from \(Y\) to \(\mathcal{L}(\mathcal{H})\). An extension \(\tau\) of \(A\) by \(C(Y) \otimes \mathcal{K}\) is said to be homogeneous if for every \(y \in Y\), the map \[\mathrm{ev}_y \circ \tau : A \rightarrow Q\] is injective, where \(\mathrm{ev}_y : Q(Y) \rightarrow Q\) denotes the evaluation map at \(y\). Let \(\mathrm{Ext}_{\mathrm{PPV}}(Y,A)\) denote the set of unitary equivalence classes of unital homogeneous extensions of \(A\) by \(C(Y) \otimes \mathcal{K}\). For a nuclear \(C^*\)-algebra \(A\), Pimsner, Popa, and Voiculescu [4] showed that \(\mathrm{Ext}_{\mathrm{PPV}}(Y,A)\) is a group. We denote the equivalence class of an extension \(\tau\) in \(\mathrm{Ext}_{\mathrm{PPV}}(Y,A)\) by \([\tau]_{su}\). Denote by \([\tau]\) the stable equivalence class of an extension \(\tau\) in the group \(\mathrm{Ext}(A,C(Y))=KK^1(A,C(Y))\).

3 Isomorphism between \(\operatorname{Ext}_{\mathrm{PPV}}(Y, A)\) and \(\operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma^2_mA)\)↩︎

In this section, we begin by proving that, under suitable assumptions on the space \(Y\) and on a unital, nuclear, and separable \(C^*\)-algebra \(A\), the groups \(\operatorname{Ext}_{\mathrm{PPV}}(Y, A)\) and \(\operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma^2_mA)\) are isomorphic. This is then used to establish the main result of the paper. Without loss of generality, we shall assume that the Hilbert space \(\mathcal{H}\) is \(\ell^2(\mathbb{N})\). Throughout this section, \(Y\) is a finite-dimensional compact metric space.

Let \(\tau\) be a unital homogeneous extension of \(A\) by \(C(Y)\otimes \mathcal{K}(\mathcal{H})\). We define \[\tilde{\tau} : A \longrightarrow Q\big(C(Y)\otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H})\big), \qquad \tilde{\tau}(a) = [\tau(a)_y\otimes p_{<m}]_{y\in Y}.\] Take \(\nu \in \Delta_m^2\). Define the extension \(\Sigma_m^2 (\tau, \nu):=\Sigma^2_m(\tilde{\tau}, \nu, [1\otimes 1 \otimes S^m])\). Equivalently, one may write \(\Sigma_m^2 (\tau, \nu)\) as the \(*\)-homomorphism given by: \[\label{sigmatau} \Sigma_m^2 (\tau, \nu) : \Sigma_m^2 A \;\longrightarrow\; Q\big(C(Y)\otimes \mathcal{K}(\mathcal{H})\otimes \mathcal{K}(\mathcal{H})\big)\tag{1}\] such that \[\Sigma_m^2 (\tau, \nu)(a \otimes p_{<m}) = \tilde{\tau}(a) = [\tau(a)_y \otimes \mathcal{P}_{<m}]_{y \in Y}, \quad \Sigma_m^2 (\tau, \nu)(1 \otimes S^m) = [1 \otimes S^m]_{y \in Y}\] and \[\Sigma_m^2 (\tau, \nu)(1 \otimes p_{ij})= [1 \otimes \nu(p_{ij})_y]_{y \in Y},for0\leq i,j \leq m-1.\] Since \(\tau\) is homogeneous, one can check that \[ev_y \circ \Sigma_m^2 (\tau, \nu)(a \otimes k)\neq 0\] for any \(a \in A\) and \(k \in \mathcal{K}\). As a consequence of Propostion 4, it follows from that the extension \(\Sigma_m^2 (\tau, \nu)\) is homogeneous. If \(\nu=0\), then we denote \(\Sigma_m^2 (\tau, \nu)\) by \(\Sigma_m^2\tau\). If \(m=1\), then we write \(\Sigma_m^2 (\tau, \nu)\) and \(\Sigma_m^2\tau\) by \(\Sigma^2 (\tau, \nu)\) and \(\Sigma^2\tau\). Observe that the extension \(\Sigma_m^2 \tau\) is nothing but the restriction of the map \(\Sigma^2 \tau\) to the subalgebra \(\Sigma_m^2A\). It follows from Lemma \(2.8\) in [5] that the map \[\beta: \operatorname{Ext}_{\mathrm{PPV}}(Y, A) \longrightarrow \operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma^2A); \quad [\tau]_{su} \;\longmapsto\; [\Sigma^2 \tau]_{su}\] is an isomorphism. Let \[i : \operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma^2A) \rightarrow \operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma_m^2A); \quad [\Sigma^2 \tau]_{su} \mapsto [\Sigma_m^2 \tau]_{su}.\] It is easy to see that \(i\) is well-defined homomorphism. Therefore, by composing it with \(\beta\), we get a homomorphism \[\Psi_m: \operatorname{Ext}_{\mathrm{PPV}}(Y, A) \longrightarrow \operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma_m^2A); \qquad \phi=i \circ \beta.\]

: Assume that \(\Psi_m([\tau]_{su})=[\Sigma_m^2\tau]_{su}=0\). Then there exists a \(*\)-homomorphism \(\overline{\Sigma^2_m\tau}: \Sigma_m^2A\rightarrow M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))\) such that \[\Sigma^2_m\tau(b)=(\pi \circ \overline{\Sigma^2_m\tau})(b) \,\,for allb \in \Sigma^2_mA.\] Let \(P_0=1\otimes 1\otimes p\). Then we have \[P_0\sim_{MVN} 1-P_0\sim_{MVN} 1.\] Thus, there exists a isometry \(T_0\in M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))\) such that \[T_0T_0^*=P_0.\] This induces the following isomorphism: \[T_0\boldsymbol{\cdot}T_0^*: M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))\rightarrow P_0M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))P_0;\quad a \mapsto T_0aT_0^*.\]

Hence we have

Note that \(P_0M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))P_0=M(C(Y) \otimes \mathcal{K}(\mathcal{H}))\otimes p\). Using this and the commutatice diagram, we get a \(*\)-homomorphism \[\tau^{\prime}: A \rightarrow M(C(Y) \otimes \mathcal{K}(\mathcal{H}); \, \tau^{\prime}(a)=\pi\circ \tau.\] This proves that \([\tau]_{su}=0\). \(\Box\)
We will show that \(\Psi_m\) is an isomorphism. To do this, we need to establish some results. Take a homogeneous extension \(\lambda: \Sigma_m^2A \rightarrow Q\big(C(Y)\otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H})\big)\). Let \(\lambda(1\otimes (S^*)^m)=V\). Since \(\lambda\) is homoeneous, hence norm full extension, it follows from Proposition \(2.5\), and Corollary \(2.7\) that there exists a unitary \(U\in M\big(C(Y)\otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H})\big)\) such that \([U]V[U^*]=[1\otimes 1 \otimes (S^*)^m]\). Since \([\lambda]_{su}=[[U]\lambda[U^*]]_{su}\), one can, without loss of generality assume that \(V=[1\otimes 1 \otimes (S^*)^m]\). Hence we have \[\lambda(1\otimes p_{<m})=[1\otimes 1 \otimes p_{<m}].\] Define \[\nu: M_m(\mathbb{C})\rightarrow Q\big(C(Y)\otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H})\big); \, p_{ij}\mapsto \lambda(1\otimes p_{ij})for all1\leq i,j \leq m-1.\] Then \(\nu \in \Delta_m^2\). Moreover, if we define \[\tilde{\tau}:A \rightarrow Q\big(C(Y)\otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H})\big); \, a \mapsto \lambda(a\otimes p_{<m})fora \in A,\] then we get \[\lambda=\Sigma^2_m(\tilde{\tau}, \nu, [1\otimes 1 \otimes (S^*)^m]).\]

: Let \(Ord\,([\Sigma^2_m(\tilde{\tau}, \nu, [1\otimes 1 \otimes (S^*)^m])]_{su})=r\). Fix \(r\) isometries \(s_1,s_2,\cdots s_r \in Q(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))\) such that \(\sum_{j=1}^r\, s_js_j^*=1\). Define \[\phi: \Sigma_m^2A \rightarrow Q(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H})); \quad \phi(b)=\sum_{i=1}^r s_i \Sigma_m^2 (\tau, \nu)(b)s_i^*.\] Then \([\phi]_{su}=r[\Sigma_m^2 (\tau, \nu)]_{su}\), and hence we have the following commutative diagram:

By restricting it to the \(C^*\)-subalgebra of \(\Sigma_m^2A\) generated by \(\{1\otimes p_{ij}:1\leq i,j \leq m\}\), we get the following commutative diagram:

This proves that \(r[\nu]=0\), hence the claim.

\(\Box\)

: Let \([\Sigma^2_m(\tilde{\tau}, \nu, [1\otimes 1 \otimes (S^*)^m])]_{su}\in \Psi_m(\operatorname{Ext}_{\mathrm{PPV}}(Y, A))\). Then there exists a homogeneous extension \(\tau^{\prime}\) such that \[\Sigma^2_m(\tilde{\tau}, \nu, [1\otimes 1 \otimes (S^*)^m]) \sim_{su} \Sigma_m^2\tau^{\prime}.\] Therefore, there exists a unitary \(U \in M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))\) such that \[[U] \Sigma^2_m(\tilde{\tau}, \nu, [1\otimes 1 \otimes (S^*)^m])(b) [U]^*= \Sigma_m^2\tau^{\prime}(b) \quadfor allb \in \Sigma_m^2A.\] Hence we have \[[U] \nu(p_{ij})[U]^*= [U] \Sigma_m^2 (\tau, \nu)(1 \otimes p_{ij}) [U]^*= \Sigma_m^2\tau^{\prime}(1 \otimes p_{ij})=1\otimes 1 \otimes p_{ij}.\] This shows that \(\nu \sim_{su} 0\), hence \([\nu]=0\). To show the forward direction, we assume that \([\nu]=0\). Let \(P_{<m}=1\otimes 1 \otimes p_{<m}\). Since \[\nu(1)=[P_{<m}],and1\otimes P_{<m}\sim_{MVN}1 \sim_{MVN}(1-P_{<m}),\] there exists a isometry \(T_{<m}\in M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))\) such that \[T_{<m}T_{<m}^*=P_{<m}.\] This induces the following isomorphism: \[T_{<m}\boldsymbol{\cdot}T_{<m}^*: M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))\rightarrow P_{<m}M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))P_{<m};\quad a \mapsto T_{<m}a T_{<m}^*.\] Hence we have

Therefore, there exists a unitary \(W\in P_{<m}M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))P_{<m}\) such that \[[W]\nu(p_{ij})[W^*]=[1\otimes 1 \otimes p_{ij}],for all0\leq i,j \leq m-1.\] Define \(V=\sum_{j=0}^\infty\, W(1\otimes 1\otimes S^m)^j\). Then \(V\) is a unitary in \(M(C(Y) \otimes \mathcal{K}(\mathcal{H}) \otimes \mathcal{K}(\mathcal{H}))\), and \[[V][1\otimes 1\otimes S^m]=[1\otimes 1\otimes S^m][V],and[V]\nu(p_{ij})[V^*]=[1\otimes 1 \otimes p_{ij}],for all0\leq i,j \leq m-1.\] Define the extension \(\Gamma=V[\Sigma_m^2 (\tau, \nu)]V^*\). Then \([\Gamma]_{su}=[\Sigma_m^2 (\tau, \nu)]_{su}\). Moreover, the extension \(\Gamma\) induces a homomorphism \(\gamma:A \rightarrow Q(C(Y) \otimes \mathcal{K}(\mathcal{H})\) such that \[\Gamma(a\otimes p)=\gamma(a) \otimes p,for alla \in A.\] It is not difficult to verify that \(\Gamma=\Sigma^2_m(\gamma)\). Therefore, we have \[[\Sigma_m^2 (\tau, \nu)]_{su}=[\Gamma]_{su} \in \Psi_m(\operatorname{Ext}_{\mathrm{PPV}}(Y, A)).\] \(\Box\)

: We prove the claim by induction on \(r\). The case \(r=0\) is clear, since \[\operatorname{Ext}_{\mathrm{PPV}}(Y,\mathbb{C}) = 0.\] Assume that the claim holds for \(r-1\). Let \([\psi]_{su} \in \operatorname{Ext}_{\mathrm{PPV}}^{\mathrm{Tor}}(Y,\Sigma_m^{2r}\mathbb{C}).\) Then \[[\psi]_{su} = [\Sigma_m^2\bigl(\tau, \, \nu, [1 \otimes 1 \otimes S^m] \bigr)]_{su},\] for some \(\nu \in \Delta_m^2\) and some homogeneous extension \(\tau\) of \(\Sigma_m^{2r-2}\mathbb{C}\) by \(C(Y) \otimes \mathcal{K}\). Since \[\operatorname{Ext}(M_m(\mathbb{C}), C(Y)) = KK^1(M_m(\mathbb{C}), C(Y))\] is a torsion-free abelian group by the Künneth Theorem (see Theorem 23.1.2, p. 234 of [6]), it follows that \[\operatorname{Ord}([\nu]) \in \{0, \infty\}.\] By Proposition 13, we have \[\operatorname{Ord}([\nu]) \leq \operatorname{Ord}\!\left([\Sigma_m^2\bigl(\tau, \, \nu, [1 \otimes 1 \otimes S^m] \bigr)]_{su}\right) < \infty.\] Hence \(\operatorname{Ord}([\nu]) =0\), and therefore by Lemma 17 \[\psi \in \Psi_m\bigl(\operatorname{Ext}_{\mathrm{PPV}}(Y,\Sigma_m^{2r-2}\mathbb{C})\bigr).\] The claim now follows from Lemma 12 and the induction hypothesis. \(\Box\)

: By the Künneth Theorem (see Theorem 23.1.2, p. 234, [6]), \[\operatorname{Ext}(M_m(\mathbb{C}), C(Y)) \cong KK^1(M_m(\mathbb{C}), C(Y))\] is a torsion-free abelian group. Hence we get \[\operatorname{Ord}([\nu]) \in \{0,\infty\}.\] Assume that \(\operatorname{Ord}([\nu]) = \infty\). Then, taking \(A = \mathbb{C}\) and using Proposition 13 and Lemma 14, we obtain \[\operatorname{Ord}([\Sigma_m^2 (0, \nu)]) = \infty, \quad \text{and} \quad \Sigma_m^2 (0, \nu) \notin \Psi_m(\operatorname{Ext}_{\mathrm{PPV}}(Y, \mathbb{C})).\] Fix \(r \in \mathbb{N}_0\). Consider the map \[\Psi_m^r: \operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma_m^{2r}A) \longrightarrow \operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma_m^{2(r+1)}A), \quad [\tau]_{su} \mapsto [\Sigma_m^2 \tau]_{su}.\] By a similar argument, \[\operatorname{Ord}([\Sigma_m^2 (0_r, \nu)]) = \infty, \quad \text{and} \quad \Sigma_m^2 (0_r, \nu) \notin \Psi_m^r\big(\operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma_m^{2r}\mathbb{C})\big).\] Thus, the elements \[\left\{ \Psi_m^{r} \circ \cdots \circ \Psi_m^{j+1} \Sigma_m^2 (0_j, \nu) \;\middle|\; 0 \le j \le r \right\}\] are \(\mathbb{Z}\)-linearly independent elements of infinite order in \(\operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma_m^{2r}\mathbb{C})\). By Proposition 9, one can identify \(\operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma_m^{2r}\mathbb{C})\) as a subgroup of \(KK^1(C(Y), \Sigma_m^{2r}\mathbb{C})\). Moreover, by the Universal Coefficient Theorem (see Theorem 23.1.1, p. 233, [6]), we have \[KK^1(\Sigma_m^{2r}\mathbb{C}, C(Y)) \cong K_0(C(Y)) \oplus K_1(C(Y)) \oplus (\mathbb{Z}/m\mathbb{Z})^{\oplus r}.\] Choose \(r\) greater than the rank of \(K_{*}(C(Y))\). Then \(\operatorname{Ext}_{\mathrm{PPV}}(Y, \Sigma_m^{2r}\mathbb{C})\) contains at least \(r+1\) \(\mathbb{Z}\)-linearly independent elements of infinite order, which is possible only if \[\operatorname{Ext}_{\mathrm{PPV}}^{\mathrm{Tor}}(Y, \Sigma_m^{2r}\mathbb{C}) \neq 0,.\] This leads to a contradiction to Lemma 15. \(\Box\)

: It suffices to show that \(\Psi_m\) is surjective, by Lemma 12. Let \(\lambda\) be a homogeneous extension of \(\Sigma_m^2 A\) by \(C(Y) \otimes \mathcal{K}\). Then \[[\lambda]_{su} = \big[\Sigma_m^2(\tilde{\tau}, \nu, [1 \otimes 1 \otimes (S^*)^m])\big]_{su}\] for some \(\nu \in \Delta_m^2\) and some homogeneous extension \(\tau\) of \(A\) by \(C(Y) \otimes \mathcal{K}\). By Lemma 16, we have \([\nu] = 0\). Therefore, Lemma 14 implies that \[\Sigma_m^2(\tilde{\tau}, \nu, [1 \otimes 1 \otimes (S^*)^m]) \in \Psi_m\big(\operatorname{Ext}_{\mathrm{PPV}}(Y, A)\big).\]

: The first part follows by a straightforward verification. For the other part, fix \(y\in Y\) and let \[J_y=\ker\big(\mathrm{ev}_y\circ\Sigma^2_m\tau\big).\] Since \(\tau\) is homogeneous, \(J_y\) is an ideal of \(\Sigma^2_m A\) with \(J_y\cap (A\otimes\mathcal{K})=\emptyset\). If \(J_y\neq\{0\}\), then there exists \(t\in\mathbb{T}\) such that \[(1\otimes S^m)-t{1\!\!1}\in J_y.\] However, we have \[\mathrm{ev}_y\circ\Sigma^2_m\tau(1\otimes S^m)=[S^m] \neq [t{1\!\!1}] \quad\text{for any }t\in\mathbb{T},\] so no such nonzero element lies in \(J_y\). Hence \(J_y=\{0\}\) for every \(y\in Y\), proving the claim. \(\Box\)

: It is a direct consequence of Theorem 17 and Proposition 18. \(\Box\)

: It follows from Proposition 18 and Theorem 5. \(\Box\)
We now establish the main result.

: Let \(1 \leq k \leq n\). In this case, it follows from the explicit description of irreducible representations given in [3] that the images of the first \(k\) generators of \(C\bigl(SO_q(2n+1)/SO_q(2n-1)\bigr)\) under the map \(\phi_{\omega_k}\) form the standard generators of \(C\bigl(S_q^{2k+1}\bigr)\), while the images of the remaining generators are zero. The claim then follows, since \[C\bigl(S_q^{2k+1}\bigr)\cong \Sigma^{2(k-1)} C(\mathbb{T}).\] To prove the claim for \(k=n+1\), assume that it holds for \(1 \leq k \leq n\). For \(m \neq 0\), we have the extension \[\xi_m:\; 0 \longrightarrow C(\mathbb{T})\otimes \mathcal{K}\longrightarrow \Sigma_m^2 C(\mathbb{T}) \xrightarrow{\sigma} C(\mathbb{T}) \longrightarrow 0, \qquad \sigma(S^m)=\boldsymbol{t}.\] To get a trivial homogeneous extension \(\xi_0\), take \(q \in \Upsilon\), where \(\Upsilon:=\{q \in \mathbb{C}: 0<|q| <1, \theta=\frac{1}{\pi}\arg{(q)} is irrational\}\). Then by [5], we have the following trivial homogeneous extension. \[\xi_0: 0 \longrightarrow C(\mathbb{T})\otimes \mathcal{K}\longrightarrow C(U_q(2)/_{\psi}\mathbb{T}) \stackrel{\vartheta}{\longrightarrow} C(\mathbb{T}) \longrightarrow 0.\] Denote \(C(U_q(2)/_{\psi}\mathbb{T})\) by \(\Sigma^2_0C(\mathbb{T})\). Then we have \[K_0(\Sigma^2_0C(\mathbb{T}))=\begin{cases} \mathbb{Z}\oplus \mathbb{Z}/m\mathbb{Z}&ifm \in \mathbb{Z}\setminus \{0\}, \cr \mathbb{Z}\oplus \mathbb{Z}&ifm=0. \cr \end{cases}\] Using Lemma \(3.4\) in [5], we get \[\mathrm{Ext}_{\mathrm{PPV}}(\mathbb{T}, C(\mathbb{T}))=\{[\xi_m]_{su}: m\in \mathbb{Z}\}.\] Since \(\Sigma_m^2 C(\mathbb{T})= \Sigma_{-m}^2 C(\mathbb{T})\), it follows that \(K_0\) is a complete invariant for the middle \(C^*\)-algebras appearing in the extensions of \(\operatorname{Ext}_{\mathrm{PPV}}(\mathbb{T}, C(\mathbb{T}))\). Then from Proposition 20, one can conclude that \(K_0\) is also a complete invariant for the middle \(C^*\)-algebras appearing in the extensions of \(\operatorname{Ext}_{\mathrm{PPV}}(\mathbb{T}, B_n)\). By Proposition 19, we obtain \[\mathrm{Ext}_{\mathrm{PPV}}(\mathbb{T}, B_n^{2n+1}) = \mathrm{Ext}_{\mathrm{PPV}}(\mathbb{T}, \Sigma^{2(n-1)} C(\mathbb{T})) = \{[\Sigma^{2(n-1)} \xi_m]: m\in \mathbb{Z}\}.\] Therefore, the middle \(C^*\)-algebras appearing in the extensions of \(\operatorname{Ext}_{\mathrm{PPV}}(\mathbb{T}, B_n)\) are given by \[\{ \Sigma^2_m\Sigma^{2(n-1)}C(\mathbb{T}): m \in \mathbb{Z}\}.\] By Theorem 5, we have \[K_0(\Sigma^2_m\Sigma^{2(n-1)}C(\mathbb{T}))=\begin{cases} \mathbb{Z}\oplus \mathbb{Z}/m\mathbb{Z}&ifm \in \mathbb{Z}\setminus \{0\}, \cr \mathbb{Z}\oplus \mathbb{Z}&ifm=0. \cr \end{cases}\] By the induction hypothesis and Lemma 6, it follows that \(B_{n+1}^{2n+1}\) appears as a middle \(C^*\)-algebra in the extensions belonging to \(\mathrm{Ext}_{\mathrm{PPV}}(\mathbb{T}, B_n^{2n+1})\). Now, using Theorem 5 and 7, we conclude that \[B_{n+1}^{2n+1}\cong \Sigma_2^2 \Sigma^{2(n-1)} C(\mathbb{T}).\] For the case \(k=n+2\), the argument is the same as given above. However, in this case one needs to use Theorem 17 or Proposition 19 to obtain the list of all extensions, and hence all the middle \(C^*\)-algebras appearing in the extensions of \(\operatorname{Ext}_{\mathrm{PPV}}(\mathbb{T}, B_{n+1})\). The remaining cases follow along similar lines. \(\Box\)

(bipul.saurabh@iitgn.ac.in, saurabhbipul2@gmail.com)
Department of Mathematics, Indian Institute of Technology, Gandhinagar, Palaj, Gandhinagar 382055, India

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