Key words: \(C^*\) bundle; Heegaard decomposition; homotopy; locally trivial; matrix bundle; quantum sphere; quantum torus; subhomogeneous

Introduction↩︎

In studying the internal structure a \(C^*\)-algebra acquires by virtue of having large central subalgebras one is naturally led to the notion of a field of \(C^*\)-algebras or \(C^*\) bundles for short ([1], [2] or [3] for the broader notion of a Banach bundle, etc.) : continuous open surjections \({\mathcal{A}} \xrightarrow[]{\pi}\mathrel{\mkern-14mu}\rightarrow X\) with \(C^*\) structures on the individual fibers \({\mathcal{A}}_x:=\pi^{-1}x\) so that the \(*\)-algebraic operations and the fiber-wise norms vary continuously in ways not difficult to make precise.

The present paper is concerned exclusively with subhomogeneous Banach/\(C^*\) bundles [4]: those with a global finite bound on the fiber dimensions \(\dim {\mathcal{A}}_x\). It is in that context that [5] studies the non-commutative branched covers introduced in [6]: subhomogeneous \(C^*\) bundles (typically over compact Hausdorff base spaces) with the additional requirement that there be

• a conditional expectation (norm-1 projection [4]) \[\text{continuous sections of {\mathcal{A}}} =: \Gamma({\mathcal{A}}) \xrightarrow[]{\quad E\quad}\mathrel{\mkern-14mu}\rightarrow C(X) := \text{continuous {\mathbb{C}}-valued functions on X};\]

• of finite index in the sense of [7]: \(KE-\mathop{\mathrm{id}}\) is a positive map for some \(K\ge 1\).

As homogeneity (i.e. fiber-dimension constancy rather than boundedness) automatically provides such expectations [5], whether or not a subhomogeneous bundle is a quantum branched cover can be construed as a gauge for how pathologically the singular (lower-than-typical-dimensional) fibers are distributed. Another such gauge is homogeneous embeddability: whether or not a subhomogeneous \(C^*\) bundle is embeddable into a homogeneous one (or into a locally trivial matrix bundle: the framing employed throughout most of the sequel). Reasons why such embeddings might not be possible are not difficult to glean:

• On the one hand, [5] provides a quantum branched cover over a one-point compactification \(X^+=X\sqcup\{*\}\) with one exceptional fiber \({\mathbb{C}}\) at the one exceptional point \(*\), typical fiber \(M_2\) elsewhere, and failing matrix-bundle embeddability because the bundle is not of finite type on \(X\): not trivializable over the members of a finite open cover.

• On the other hand, in otherwise very tame topological situations a bundle with one exceptional \({\mathbb{C}}^2\) fiber and \(M_3\) generic fiber fails matrix-bundle embeddability because it fails to admit a faithful tracial expectation ([8], [9]).

It becomes natural at this stage to ask whether, such obstacles being absent, matrix-bundle embeddability can fail for a quantum branched cover \({\mathcal{A}} \xrightarrow[]{}\mathrel{\mkern-14mu}\rightarrow X\) for reasons more directly traceable to topological-algebraic properties of the locally trivial bundles obtained by restricting \({\mathcal{A}}\) to its strata: loci of constant fiber isomorphism class. With that object in view we examine subhomogeneous \(C^*\) bundles attached to various quantum-manifold function algebras. Recall the quantum (or non-commutative) tori \({\mathbb{T}}_{\theta}^n\) and spheres \({\mathbb{S}}_{\theta}^{2n-1}\) of, say, [10] and [11] respectively: defined dually by describing their associated (non-commutative) \(C^*\) complex-function algebras by generators/relations: for \(\theta=-\theta^t\in M_n({\mathbb{R}})\), \[\begin{align} A^n_{\theta}:=C({\mathbb{T}}^n_{\theta}) &:= C^*\Braket{\text{unitary }U_{1\le i\le n} \big| \forall(i,j)\left(U_j U_i = e^{2\pi i \theta_{ij}}U_iU_j\right)}\\ C^n_{\theta}:=C({\mathbb{S}}^{2n-1}_{\theta}) &:= C^*\Braket{\text{normal }T_{1\le i\le n} \big| \forall(i,j)\left(T_j T_i= e^{2\pi i \theta_{ij}}T_iT_j\right) ,\quad \sum_i T_i^*T_i=1}. \end{align}\]

The second displayed family, it turns out, provides examples “in nature” (for rational deformation parameters \(\theta\)) of precisely the type alluded to above. The main result to that effect reads as follows.

The proof reduces fairly quickly via extant literature to the case \(n=2\) (3-spheres), where some of the algebraic-topology machinery alluded to above becomes serviceable.

1 Quantum spheres and singular solid non-commutative tori↩︎

For rational \(\theta=\left(\theta_{ij}=\frac{p_{ij}}{q_{ij}}\right)_{i,j}\in M_n({\mathbb{Q}})\) we have ([12] or [13] for \(n=2\), [14], say, in general) \[\label{eq:a46is46end46e} A^n_{\theta} \cong \mathop{\mathrm{\mathrm{End}}}({\mathcal{E}}) = \Gamma\left({\mathcal{E}}nd \left({\mathcal{E}}\right)\right) = \Gamma\left({\mathcal{E}}\otimes {\mathcal{E}}^*\right)\tag{2}\] for a projectively flat [15] vector bundle \({\mathcal{E}}\) over the classical torus \({\mathbb{T}}^n\) recoverable as the spectrum \(\mathop{\mathrm{\mathrm{Max}}}Z(A^n_{\theta})\) of the center \(Z(A^n_{\theta})\). The rank \(r=r_{\theta}\) of \({\mathcal{E}}\) is given explicitly in [14], and is precisely the lowest-terms denominator of \(\theta_{12}=\frac{p}{q}\) for \(n=2\). The identification \[\label{eq:cntheta} C^n_{\theta} \ni T_i \xmapsto{\quad} t_i U_i \in C\left({\mathbb{S}}^{n-1}_{\ge 0},A^n_{\theta}\right) ,\quad {\mathbb{S}}^{n-1}_{\ge 0} := \left\{\left(t_1,\cdots,t_n\right)\in {\mathbb{R}}^n_{\ge 0}\;:\;\sum t_i^2=1\right\}\tag{3}\] is an embedding [11] giving \(C^n_{\theta}\) its own realization as \(\Gamma({\mathcal{A}})\) for a subhomogeneous \(C^*\) bundle \({\mathcal{A}} \xrightarrow[]{}\mathrel{\mkern-14mu}\rightarrow \mathop{\mathrm{\mathrm{Max}}}Z\left(C^n_{\theta}\right)\) with generic fiber \(M_{r_{\theta}}\). For \(n=2\) and rational \[\theta= \begin{pmatrix} 0&\theta_{12}\\ -\theta_{12}&0 \end{pmatrix} ,\quad \theta_{12}=\frac{p}{q} ,\quad p\in {\mathbb{Z}},\;q\in {\mathbb{Z}}_{>0},\;\gcd(p,q)=1\] the situation is particularly pleasant [16]:

- *the simple connectivity of ${\mathbb{S}}^3$, ensuring
  structure-group reduction to *connected* Lie groups;*

- *reduction to maximal compact subgroups therefrom (as recalled in
  [@2605.10752v1] for instance);*

- *and the fact [@btd_lie_1995] that compact Lie groups (hence also
  all Lie groups) have vanishing $\pi_2$.*

*Indeed: the *Dixmier-Douady class* [@gbvf_ncg]
$\alpha({\mathcal{A}})\in H^3({\mathbb{T}}^2,{\mathbb{Z}})$ vanishes
for obvious dimension reasons, hence [@hjjm_bdle] the existence of
${\mathcal{E}}$. As to the uniqueness claim, it is a consequence of
the *fiber-sequence* fragment
$$B{\mathbb{S}}^1 \xrightarrow{\quad} BU(\bullet) \xrightarrow{\quad} BPU(\bullet)$$
of classifying spaces attached to
$$\{1\} \to {\mathbb{S}}^1 \lhook\joinrel\xrightarrow{\quad} U(\bullet) \xrightarrow[]{\quad}\mathrel{\mkern-14mu}\rightarrow PU(\bullet) \to \{1\}:$$
The right-hand map classifies
${\mathcal{E}}\mapsto {\mathcal{E}}\otimes {\mathcal{E}}^*$ while
$B{\mathbb{S}}^1$ classifies line bundles. Alternatively:*

- *Two $q\times q$-matrix bundles over a 2-torus are isomorphic
  precisely when their corresponding characteristic classes (the
  $\beta_q\in H^2({\mathbb{T}}^2,{\mathbb{Z}}/q)$ of [@hjjm_bdle])
  are equal; this is noted in broader generality as part of
  [@2509.10812v2].*

- *This in turn means that
  ${\mathcal{E}}\otimes {\mathcal{E}}^*\cong {\mathcal{F}}\otimes {\mathcal{F}}^*$
  if and only if the first Chern classes of the two rank-$q$ vector
  bundles are equal modulo $q$.*

- *Finally, the complete topological characterization [@MR1423157]
  of a vector bundle over a surface by Chern class and rank renders
  this equivalent to
  ${\mathcal{F}}\cong {\mathcal{E}}\otimes {\mathcal{L}}$ for a line
  bundle ${\mathcal{L}}$.*

We retain the notation employed in 1, and supplement it with subscripts \(F\subseteq [0,1]\) for both \({\mathbb{S}}^3\) and the bundle \({\mathcal{S}}\) to indicate \(\eta^{-1}F\) (see ?? ) and the restriction \({\mathcal{S}}|_{\eta^{-1}F}\) respectively. The elements \(U_i^q\in A^2_{\theta}\) can be regarded as coordinates on any of the tori \({\mathbb{T}}^2_t\), \(t\in (0,1)\), with \(U^q_{t+1}\) also acting as a respective coordinate along the exceptional circle \({\mathbb{S}}^1_{t\in \{0,1\}}\).

Recall that per the bundles-on-spheres classification of [18] bundles over \({\mathbb{S}}^1\) (hence also the homotopy-equivalent solid tori) with path-connected structure group are automatically trivial. This applies in particular to vector and matrix bundles, their structure groups being unitary and projective unitary respectively.

It will be profitable to isolate a concrete technical embeddability criterion. We first need a bit of notation referring to shifted diagonal matrices.

Throughout the ensuing discussion \(\mathop{\mathrm{Ad}}_T\) denotes conjugation \(T\bullet T^{-1}\). Note also, in preparation for stating 5, that embeddings \(M_q\le M_n\) (and hence of \(M_q\) bundles into \(M_n\) bundles) exist only if \(q|n\). For a reduced rational \(\frac{p}{q}\) (featuring prominently in the sequel as the torus/sphere-deformation parameter) fix \[\label{eq:diag46subdiag} D:=\mathop{\mathrm{\mathrm{diag}}}_0\left(\zeta^i\right)_{i=0}^{q-1} ,\;\zeta := e\left(\frac{p}{q}\right) := \exp\left(2\pi i\cdot \frac{p}{q}\right).\tag{5}\] For a finite-dimensional \(C^*\)-algebra \(A\) we write \(\mathop{\mathrm{\mathrm{Emb}}}_{A,M_n}\) for the space of unital \(C^*\)-embeddings \(A\le M_n\) and \(\mathop{\mathrm{\mathrm{Emb}}}^{=}_{A,M_n}\) for only those unital embeddings giving all simple factors of \(A\) equal multiplicities; \(\mathop{\mathrm{\mathrm{Emb}}}^=_{M_q,M_n}\) is non-empty, then, precisely when \(q|n\).

We record a consequence, weaker than 2 and to be superseded by the latter later.

5 can now be leveraged into a sharper equivalence criterion.

As the torus bundles of concern in 2 are obtained by restricting the sphere-based ones of 1, an immediate consequence of the theorem is its analogue for quantum 3-spheres.

Department of Mathematics, University at Buffalo

Buffalo, NY 14260-2900, USA

E-mail address: achirvas@buffalo.edu

References↩︎