I’m getting sick of the highly nested threading above, so I’ll reply to Jamie’s comment about 2-C*-algebras here. Jamie wrote:

Hmm… I’m pretty sure I haven’t seen a definition of a 2-C*-algebra before.

Neither have I.

I know what C*-algebras and H*-algebras are, I know what C*-categories and H*-categories are, and I know what
2-H*-algebras are. Some of the definitions are conveniently listed in a post by Urs. I’ve never tried defining 2-C*-algebras, but someone should!

To understand 2-C*-algebras we really should understand infinite-dimensional 2-Hilbert spaces. I defined finite-dimensional 2-Hilbert spaces a while ago, but I’m still just gradually working my way towards understanding the infinite-dimensional ones. The higher Hilbert spaces discussed in my paper with Baratin, Freidel and Wise are *not* the final correct notion of 2-Hilbert space, but they’re close.

Do you not think 2-H*-algebras will be good enough for the hypothesis?

No, not for the full-fledged version of the Gelfan’d–Naimark Hypothesis mentioned above.

We can see this already one notch down. A finite-dimensional H*-algebra is almost the same thing as a finite-dimensional C*-algebra — there are functors going back and forth between H*-Alg and C*-Alg, and they *almost* form an equivalence.

(The only difference is that we don’t require $\| 1 \| = 1$ in an H*-algebra, while we do in a C*-algebra. This leads to some subtle issues that would be fun to explore elsewhere… I believe Bruce Bartlett has thought about these a bit! But, they’re not the main point here.)

On the other hand, infinite-dimensional C*-algebras are far more general than infinite-dimensional H*-algebras.

So, there’s a nice Gelfan’d–Naimark theorem for commutative C*-algebras, saying any of these is the algebra of continuous complex functions on a compact Hausdorff space, and conversely.

But, this isn’t true for commutative H*-algebras.

However, we can say any *finite-dimensional* commutative H*-algebra is the algebra of complex functions on a *finite* measure space, and conversely. This is a ‘baby’ version of the Gelfan’d–Naimark theorem.

(Measure space? Well, this is where the subtle parenthetical remark above rears its ugly head.)

There is also a theorem of Gelfan’d–Naimark type that applies to *arbitrary* commutative H*-algebras — but it’s still sort of a ‘baby’ version the one for commutative C*-algebras.

Given all this, we’d naturally expect the Gelfan’d–Naimark theorem for symmetric 2-H*-algebras to be just a baby version of the version for 2-C*-algebras (whatever those turn out to be).

And as you know, I proved this baby version in HDA2.

Or is a 2-C*-algebra the same thing as a 2-H*-algebra?

We surely don’t want that.

I don’t see how the difference between H*-algebras and C*-algebras categorifies!

Neither do I, completely, but the difference between H*-categories and C*-categories is very instructive.

(Unfortunately, the word “C*-category” seems to be ungooglable — try papers by Michael Müger and others if you want to see the definition.)

## Re: Classical vs Quantum Computation (Week 2)

The “unit” string-diagram laws on page 21 can be paired with an “associativity” string-diagram law stating that (Δ × id) and (id × Δ) become equal when preceded by Δ (both composites simply

tripleinformation). Here is a cheap ASCII drawing:These laws should be all that you need to manipulate Δ and ! in string diagrams mechanically.

It would be nice if (in general closed monoidal categories, not just in compact ones) the evaluation map

X⊗ hom(X,Y) →Y(see the final page) could be drawn invisibly (like the other natural maps in a closed monoidal category). You would need to have the “ribbon” hom(X,Y) open up; perhaps the correct metaphor is azipper?