Having completed my MSc on Random Walks on Finite Groups, it’s time to go quantum! Quantum here refers primarily to non-commutative geometry. In studying a group $G$ (or some other geometry), often a study of the algebra of complex functions on the group, $F(G)$ (or some similar object depending on the class of group), can tell us everything about the group – we can reconstruct the underlying structure from the algebra of functions on it. In particular, the algebra of functions is a commutative algebra that can encodes the group axioms in a certain way.

Forget about the group for a minute, and consider an algebra $\mathcal{A}$ that appears to encode the group axioms in the same way (exactly as the introduction in http://arxiv4.library.cornell.edu/PS_cache/q-alg/pdf/9704/9704002v2.pdf). This algebra is then called an algebra of functions on a quantum group, where a quantum group is an abstract object whose algebra of functions is given by $\mathcal{A}$. Of course, in this setting, the algebra of functions is no longer commutative hence the terminology.

Franz and Gohm http://www.springerlink.com/content/p30834372420lp4u/fulltext.pdf present a formulation of a random walk on a finite quantum group. As we study an algebra of functions rather than the underlying space itself, a number of generalised concepts come into play: especially a quantum theory of probability.

It would be particularly interesting for me to study in this formulation a random walk on a group – there is a folklore theorem that if we restrict to a commutative algebra/ quantum group, a random walk on a finite group may be recovered. We may also want to ask the question: what does an Abelian quantum group look like – do there exist truly non-commutative quantum groups that encode a commutativity in the underlying space. I see no reason why not.

As a serious course of study, having seen the proliferation of tensor products in Franz & Gohm’s formulation, I began a study of Tensor Products. Wegge-Olsen http://books.google.ie/books?id=EVOjQgAACAAJ&dq=Wegge%20Olsen%20K%20Theory&source=gbs_book_other_versions has a good-sized appendix devoted to the topic. Also it is seen that quantum groups are classes of $C^\star$-Algebras and thus I had a look at Conway also http://books.google.ie/books?id=ix4P1e6AkeIC&printsec=frontcover&dq=Conway+A+course+in+Functional+Analysis&hl=en&ei=K-OyTMyHHMK4jAeQvPVx&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDQQ6AEwAA#v=onepage&q&f=false.

Both studies collapsed when there were remarks made to extensions, embeddings, and in the case of Tensor products, suitable norms. This meant a review – and an extension of – my knowledge of functional analysis. I have done a module in Functional Analysis http://www.ucc.ie/modules/descriptions/MA.html#MA4052, and so I chose to go through Belton’s notes http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf to extend my knowledge at this point.

I have been going through the notes slowly and spending 15 mins per exercise trying to come up with a solution, after which I consult the solution at the back of the notes. At present I have studied up to Tychonov’s Theorem, p.39. I haven’t had great success with the exercises – any time I get one out the solution is described as easy! Hopefully however the theory and theorems sink in and I get familiar with the results and applications of this area. Shortly I will be getting onto the chapter on algebras and I have never properly studied them before.

A particular challenge for me is in developing my mathematical maturity. In the first instance, almost everything I’ve done up to this point, including my MSc thesis, has been about objects I have no problem visualising. If I ever was confused by some topic, I could visualise or write down an easy example and see how the theorem applies, etc. For quantum groups however I must acquire much more knowledge and work hard so that properties of these more abstract objects become second-nature to me – I may no longer always have a good ituitive idea of how an object works.

Secondly I must become more careful and precise. In my MSc thesis I was doing things like writing down the norm of the stochastic operator, $\|P\|$, without any reference to which normed spaces it was acting between. The question of norm is a vital and precise one in the study of $C^\star$-Algebras and I will need to be more sophisticated and careful.

As an example of where I can be very sloppy, consider the following exercise from Belton (2.4).

Prove that no infinite-dimensional Banach space $E$ has a countable Hamel basis (where a Hamel basis is a linearly independent set $S$ such that every vector in $E$ is a finite linear combination of elements of $S$).

My attempt went as follows: Assume that $E$ is a Banach Space with a countable Hamel basis $S=\{s_i\}_{i\geq 1}$ be a Hamel basis for $E$. Transform by $S^\prime=\{s_i^\prime\}_{i\geq 1}=\{\frac{s_i}{\|s_i\|2^i}\}_{i\geq 1}$. Consider the series

$\sum_{i=1}^\infty s_i^\prime$

Now

$\sum_{i=1}^\infty\|s_i^\prime\|=\sum_{i=1}^\infty\frac{1}{2^i}$

so the series is absolutely convergent. As $E$ is a Banach space, this implies that the sum of the series, say $s$, is in $E$. If the $s_i^\prime$ are linearly independent, then $s=\sum_{i=1}^\infty s_i^\prime$ uniquely so $s$ may not be written as a finite linear combination from $S$.

This was my solution, but I realised quickly that it is wrong. Linear independence is defined with respect to finite linear combinations, not infinite. I could probably prove a few things with the shoddy argument shown here!

What’s probably worse is doing an exercise and not knowing whether it is correct or not. Another example, Belton exercise 4.2:

Let $M$ be a finite-dimensional subspace of the normed space $X$ and let $N$ be a closed subspace of $X$ such that $X = M\oplus N$. Prove that if $\phi_0$ is a linear functional on $M$ then

$\phi: M \oplus N\rightarrow \mathbb{F}; m + n \mapsto \phi_0(m),\,\,\forall\,m\in M\,,\,\,n\in N$

is an element of the dual space $X^\star$.

Solution: Clearly $\phi$ is linear. Note that $X/N$ is isomorphic to $M$ as normed vector spaces. Let $\psi:X/N\rightarrow M$ be an isometric isomporphism and let $\pi:X\rightarrow X/N$ be the quotient map. Thence $\phi=\phi_0\circ \psi\circ \pi$. These maps are all continuous hence so is $\phi$. Also:

$\|\phi(x)\|\leq \|\phi_0\|\|\psi\|\|\pi\|\|x\|=\|\phi_0\|\|x\|$

so $\phi\in X^\star$.

I have no idea if this is correct or not!