A recent talk in the noncommutative geometry seminar here, was by Farzad Fathizadeh. He was talking about a few ideas – the main part of the talk being about how to construct the Dixmier-Douady invariant, which is related to the question of whether or not you can put a spin structure on some manifold. It’s also related to a lof other things I want to figure out anyway for longstanding reasons. Indeed, Dixmier is one of the big early names behind the theory of fields of Hilbert spaces, which are used in Crane and Yetter’s “Measurable Categories”, which are a sort of infinite dimensional analog of the 2-vector spaces I’ve been talking about. (Actually, 2-Hilbert spaces, since that structure starts to look more important there).

Since I’ve started thinking about infinite dimensional 2-Hilbert spaces again I thought I’d check it out. It turned out to be somewhat related, but not very deeply. However, precisely because it’s related to things I’ve yet to figure out, I’m going to give a superficial gloss here, and later maybe try to say something more detailed. I should be giving a talk to our group soon about various aspects of 2-Hilbert spaces, so I’ll post more when I get to that.

The second part of the talk had a nice exposition of Morita equivalence, which was a notion that got a lot of use in the things people were talking about at Groupoidfest. I had heard about this concept before, but never quite got the hang of it until now, so here’s a quick little explanation for the record. There are two ways of describing Morita equivalence, and the content of Morita’s theorem is that the two definitions amount to the same thing.

One definition says that two algebras $A$ and $B$ are equivalent if the categories of modules over them, $Mod(A)$ and $Mod(B)$ are equivalent as categories. The other says that $A$ and $B$ are equivalent if there is an $A-B$-bimodule, $\mathcal{F}$, and a $B-A$-bimodule, $\mathcal{G}$ with the properties that:

$\mathcal{F} \otimes_{B} \mathcal{G} \cong A$ (as an $A-A$-bimodule)

and

$\mathcal{G} \otimes_{A} \mathcal{F} \cong B$ (as a $B-B$-bimodule)

Where, if you’re unclear, an $A-B$-bimodule is a set where the algebra $A$ acts on the left, and the algebra $B$ acts on the right, with a compatibility condition that looks like associativity: $(ax)b = a(xb)$.

It shouldn’t be too hard to see that the second definition implies the first: given a (left) $A$-module $M$, you can turn it into a (left) $B$ module by taking $\mathcal{G} \otimes_{A} M$, and given a $B$-module, you turn it into an $A$-module by similarly tensoring over $B$ on the left with $\mathcal{F}$. Doing both operations gives you $\mathcal{F} \otimes_{B} \mathcal{G} \otimes_A M$. The assumptions on $\latex \mathcal{F}$ and $\mathcal{G}$ mean that this is equivalent to $A \otimes_A M \cong M$. The same (switching $\mathcal{F}$, \mathcal{G}\$ and $A,B$) goes for a $B$-module. So this is the equivalence from the first definition.

The hard part of Morita’s theorem is that any time you have an equivalence, you can represent it in terms of some bimodules $\mathcal{F}$ and $\mathcal{G}$.

(This is related to the fact that there’s a bicategory structure for algebras in which the morphisms from $A$ to $B$ are $A-B$-bimodules, and the 2-morphisms are bimodule homomorphisms. Composition works by exactly the kind of tensoring above. In fact, there’s a pseudocategory of rings, which has a horizontal bicategory like the one I just described, and a vertical category (a strict category this time) where the morphisms are ordinary ring homomorphisms. Then there are some square-shaped 2-cells, as well. When I had recently put out that paper about double bicategories of cobordisms, and ran into Peter May at the Fields Institute, he suggested I should look into this stuff more. At the time I didn’t quite see why people wanted these structures that were weak in only one direction – but it’s a little clearer now.)

Anyway, on the subject of Morita equivalence, Masoud Khalkhali pointed out that there’s a fairly easy example which makes a connection to the “bra-ket notation” which Dirac introduced to physics. This is the example that says an algebra $A = k$ is Morita equivalent to any of the associated algebras of square matrices $B = M_n(k)$. Then the bimodules that achieve the equivalence are $\mathcal{F} = k^n$ (as row vectors, acted on by field elements on the left, matrices on the right), and $\mathcal{G} = (k^n)^*$ (column vectors – vice versa).

You can think of a row vector as a “bra” $\langle \phi | = (\phi_1, \dots ,\phi_n)$, which is acted on by operators on the right, and a column vector as a “ket” $| \psi \rangle = (\psi_1, \dots , \psi_n)^T$ acted on by operators on the left. In physics, a “ket” designates a state you might measure, and a “bra” a state you might set a system up in. Then you can have expressions like $\langle \phi | \psi \rangle$, denoting an inner product, or $\langle \phi | A | \psi \rangle$, which gives a complex number. This denotes the “transition amplitude” for a system set up in the state $\langle \phi |$, which has evolved according to (or been acted on by) the operator $A$ to be measured in state $| \psi \rangle$. If $A$ is unitary, you tend to think of this as something like time evolution – if the operator is self-adjoint, it can be interpreted as a physical observable, in which case $\langle \psi | A | \psi \rangle$ is the expected value of the observable.

Part of the point of the balanced Dirac notation is that the operator can be thought of as acting on either the bra or the ket – you can move $A$ across the middle. So actually $\langle \phi | A | \psi \rangle$ is a particular representative of some element of $\mathcal{F} \otimes_B \mathcal{G}$ according to our definitions above (with $k = \mathbb{C}$). So this is equivalent to $\mathbb{C}$.

Similarly, expressions like $| \psi \rangle \langle \phi |$ denote particular linear operators (if you like, you could put a scalar multiple in the middle!) That particular one is the operator which takes a (unit) state vector $| \phi \rangle$ and spits out $| \psi \rangle$, and ignores anything orthogonal to $| \phi \rangle$. Any linear operator is generated by ones like this, so $B = M_n(\mathbb{C}) \cong \mathcal{G} \otimes_A \mathcal{F}$ in the notation above.

Anyway, this is a rather nice aid to grasping Morita equivalence. It’s also a special case of what was really being discussed – algebras (and modules) which look like these pointwise, but actually consist of bundles whose fibres are algebras (or modules, or what have you) that look like the above. I don’t really understand what these are good for yet, but they seem great aesthetically.

At some later point, I’ll come back to the stuff I hinted at above about 2-Hilbert spaces. This should be important in linking the extended TQFT stuff I discussed in earlier posts with “actual” 3D quantum gravity.