The existence of $\Spin^c$ structures

Suppose $n = 2m + 1 = \dim M$ is odd. Then the fibres of $B$ are semisimple but not simple: $\bCl(T_x^* M) \isom M_{2^m}(\bC) \oplus M_{2^m}(\bC)$. We shall restrict to the even subalgebras, $\bCl^0(T_x^*M) \isom M_{2^m}(\bC)$, by demanding that $c(\ga)$ act as the identity in all cases. Then we may adopt the convention that

$$c(\ka) := c(\ka\ga) \quad\text{when $\ka$ is odd}.$$

Notice here that $\ka\ga$ is even; and $c(\ga) = c(\ga^2) = +1$ is required for consistency of this rule.

We take $A = C(M)$, but for $B$ we now take

$$B := \begin{cases} \Ga(M, \bCl(T^*M)), & \text{if $\dim M$... ...M, \bCl^0(T^*M)), & \text{if $\dim M$ is odd}. \end{cases}$$ (6)

The fibres of these bundles are central simple algebras of finite dimension $2^{2m}$ in all cases.

We classify the algebras $B$ as follows. Taking

$$\ul{B} := \begin{cases} \set{B_x = \bCl(T_x^* M) : x \in M... ..._x^* M) : x \in M}, & \text{if $\dim M$ is odd} \end{cases}$$

to be the collection of fibres, we can say that $\ul{B}$ is a ``continuous field of simple matrix algebras'', which moreover is locally trivial. There is an invariant

$$\delta(\ul{B}) \in \rH^3(M;\bZ)$$

for such fields, found by Karrer [Kar] and in more generality --allowing the compact operators $\sK$ as an infinite-dimensional simple matrix algebra-- by Dixmier and Douady [Dix].

Here is a (rather pedestrian) sketch of how $\delta(\ul{B})$ is constructed:

If $x \in M$, take $p_x \in B_x$ to be a projector of rank one, that is,

$$p_x = p_x^* = p_x^2 \words{and} \tr p_x = 1.$$

On the left ideal $S_x := B_x p_x$, we introduce a hermitian scalar product

$$\braket{a_x p_x}{b_x p_x} := \tr(p_x a_x^* b_x p_x ).$$ (7)

Notice that the recipe

$$\ketbra{a_x p_x}{b_x p_x} : c_x p_x \mapsto (a_x p_x)(b_x p_x)^*(c_x p_x) = (a_x p_x b_x^*)(c_x p_x)$$

identifies $\sL(S_x)$ --or $\sK(S_x)$ in the infinite-dimensional case-- with $B_x$, since the two-sided ideal $\spn\set{a_x p_x b_x^* : a_x, b_x \in B_x}$ equals $B_x$ by simplicity.

By local triviality, this can be done locally with varying $x$. If $\{U_i\}$ is a ``good'' open cover of $M$, we get local fields $\ul{S}_i = \set{S_{i,x} : x \in U_i}$ with isomorphisms $\ul{\th}_i \: \sL(\ul{S}_i) \to \ul{B}\bigr\vert _{U_i}$ of fields of simple $C^*$-algebras. On nonempty intersections $U_{ij} := U_i \cap U_j$, we get $*$-algebra isomorphisms $\ul{\th}_i^{-1} \ul{\th}_j \: \sL(\ul{S}_j) \to \sL(\ul{S}_i)$, so there are fields of unitary maps $\ul{u}_{ij} \: \ul{S}_j \to \ul{S}_i$ such that $\ul{\th}_i^{-1}\ul{\th}_j = \ul{u}_{ij}(\cdot)\ul{u}_{ij}^{-1}$.

On $U_{ijk} := U_i \cap U_j \cap U_k$, we see that $(\Ad \ul{u}_{ij})(\Ad \ul{u}_{jk}) = \Ad \ul{u}_{ik}$, and so

$$\ul{u}_{ij} \ul{u}_{jk} = \ul{\la}_{ijk} \ul{u}_{ik},$$

where $\ul{\la}_{ijk} \: U_{ijk} \to \bT$ are scalar maps. We may now check that $\ul{\la}_{jkl} \ul{\la}_{ikl}^{-1} \ul{\la}_{ijl} = \ul{\la}_{ijk}$ on $U_{ijkl}$. Thus $\ul{\la}$ is a Cech 2-cocycle, and its Cech cohomology class lies in $\check\rH{}^2(M;\bT) \isom \rH^3(M;\bZ)$. We may go one more step in order to exhibit this isomorphism: if we write $\ul{\la}_{ijk} = \exp(2\pi i \ul{f}_{ijk})$ --we can take logarithms since $U_{ijk}$ is simply connected-- then

$$a_{ijkl} := \ul{f}_{ijk} - \ul{f}_{ijl} + \ul{f}_{ikl} - \ul{f}_{jkl}$$

takes values in $\bZ$ (and since each $U_{ijkl}$ is connected, these will be constant functions); thus, these $a_{ijkl}$ form a $\bZ$-valued 3-cocycle, $a$. Finally, one may check that its class $[a] \in \rH^3(M;\bZ)$ is independent of all choices made so far. We define $\delta(\ul{B}) := [a]$, which is called the Dixmier-Douady class of $\ul{B}$.

Suppose now that the Hilbert spaces $S_x \isom \bC^{2^m}$ can be chosen globally for $x \in M$ --not just locally for $x \in U_i$-- that is, they are fibres of a vector bundle $S \to M$ (that may b gifted with a Hermitian metric) such that $\sL(S_x) \isom B_x$, for $x \in M$, via a single field of isomorphisms $\ul{\th} \: \ul{\sL(S)} \to \ul{B}$ such that $\ul{\th}_i = \ul{\th}\bigr\vert _{U_i}$ for each $U_i$. Then $\ul{u}_{ij} = \ul{\th}_i^{-1} \ul{\th}_j = \id$ over $U_{ij}$, and so $\ul{\la}_{ijk} = 1$ over $U_{ijk}$, and $a_{ijkl} = 0$ over each $U_{ijkl}$; hence $\delta(\ul{B}) = [a] = 0$ in $\rH^3(M;\bZ)$.

Conversely if $\delta(\ul{B}) = 0$, so that $[\ul{\la}]$ is trivial in $\check\rH{}^2(M;\bT)$, i.e., $\ul{\la}$ is a 2-coboundary, then there are maps $\ul{\nu}_{ij} \: U_{ij} \to \bT$ such that $\ul{\la}_{ijk} = \ul{\nu}_{ij} \ul{\nu}_{ik}^{-1} \ul{\nu}_{jk}$ on $U_{ijk}$. Setting $\ul{v}_{ij} := \ul{\nu}_{ij}^{-1} \ul{u}_{ij}$, we get local fields of unitaries such that $\ul{v}_{ij} \ul{v}_{jk} = \ul{v}_{ik}$ on each $U_{ijk}$. These $\ul{v}_{ij} \: \ul{S}_{j} \to \ul{S}_{i}$ are therefore transition functions for a (Hermitian) vector bundle $S \to M$ such that $\ul{S}\bigr\vert _{U_i} \isom \ul{S}_{i}$ for each $U_i$. Let $\sS := \Ga(M, S)$ denote the $A$-module of sections of this bundle. Now the pointwise isomorphisms $B_x \isom \End S_x$, for each $x \in M$, imply that $B \isom \End_A S$ as $A$-modules, and indeed as $C^*$-algebras. We summarize all this in the following Proposition.