Matrix Bundles and Operator Algebras Over a Finitely Bordered Riemann Surface

Abstract

This note presents an analysis of a class of operator algebras constructed as cross-sectional algebras of flat holomorphic matrix bundles over a finitely bordered Riemann surface. These algebras are partly inspired by the bundle shifts of Abrahamse and Douglas. The first objective is to understand the boundary representations of the containing $C^*$-algebra, i.e. Arveson’s noncommutative Choquet boundary for each of our operator algebras. The boundary representations of our operator algebras for their containing $C^*$-algebras are calculated, and it is shown that they correspond to evaluations on the boundary of the Riemann surface. Secondly, we show that our algebras are Azumaya algebras, the algebraic analogues of n-homogeneous $C^*$-algebras.

Appendix

For a $C^\infty$ real manifold [23, Sect. 2], or a Riemann surface [22, Sect. 6], one can define a flat vector bundle as a vector bundle which has a coordinate representative with locally constant transition functions. This is equivalent to the following perspective. Let $\underline{GL_n(\mathbb {C})}$ be the constant sheaf over S, $\mathcal {GL}_n(\mathbb {C})_h$ the sheaf of germs of holomorphic $GL_n(\mathbb {C})$-valued functions, $i : {\underline{GL_n(\mathbb {C})}} \rightarrow \mathcal {GL}_n(\mathbb {C})_h$ the inclusion of sheaves, and $i_*: H^1(S,{\underline{GL_n(\mathbb {C})}}) \rightarrow H^1(S,\mathcal {GL}_n(\mathbb {C})_h)$ the induced map. Then a vector bundle with locally constant transition functions can be regarded as an element in the image of $i_*$ (See, for example, [22, Sect. 6]).

Proof of Lemma 2.3

Let’s first consider the holomorphic case. In [19, Lem. 27], Gunning gives a 1–1 correspondence between the sets $\text {Hom}(\pi _1(S),U_n)/U_n$ and $H^1(S,{\underline{U_n(\mathbb {C})}})$. The same argument follows for the group $PU_n(\mathbb {C})$. Gunning’s proof quite clearly and visually connects an element of $\text {Hom}(\pi _1(S),G)/G$ with a coordinate bundle associated to a particular open cover $\mathcal {U}$ of S. The continuous case is addressed in [12, Thm. 13.9]. For a discrete group G, Steenrod outlines the 1–1 correspondence between elements of $\text {Hom}(\pi _1(S),G)/G$ and elements in $H^1(S,{\underline{G}})$, for any topological space S that is locally compact, Hausdorff, second-countable, path-connected, locally path-connected, and semilocally simply connected. $\square$

Proof of Lemma 2.4

The map $i_* : H^1(S,{\underline{PU_n (\mathbb {C})}}) \rightarrow H^1(S,\mathcal {PU}_n(\mathbb {C})_h)$ will be injective (in fact, an isomorphism), since an equivalence between coordinate bundles, viewed as elements in $Z^1(S,\mathcal {PU}_n(\mathbb {C})_h)$, must be implemented by a cochain $(\lambda _U)_\mathcal {U}$ in which each $\lambda _U$ is holomorphic and $PU_n(\mathbb {C})$-valued—that is, locally constant. $\square$

Proof of Lemma 2.5

We follow the proof in [17, pg. 306], substituting Lemma 2.4 for the analogous correspondence for vector bundles used by Widom. $\square$