**This is intended to be the subject of a short postgraduate talk in UCC. At times there will be little attempt at rigour — mostly I am just concerned with ideas, motivation and giving a flavour of the philosophy. Also it is fully possible that I have got it completely wrong in my interpretation!**

### Introduction

It is a theme in mathematics that geometry and algebra are dual:

Arguably this theme began when Descartes began to answer questions about synthetic geometry using the (largely) algebraic methods of coordinate geometry. Since then this duality has been extended and refined to consider:

Here a space is a set of points with some additional structure, and the idea is that for a given space, there will be a canonical algebra of functions on the space. For example, given a compact, Hausdorff topological space , the canonical algebra of functions is — the continuous functions on .

In this talk we will be mostly concerned with what properties of the space we can recover from the algebra of functions on it — in the above example we could in fact recover everything from the canonical algebra of functions. In this talk mostly we will not necessarily be talking about the canonical algebra of functions.

## Cardinality

Let be a set of three distinct points in space. Now consider the space of complex valued functions . Now to define , all we must do is choose three complex numbers:

; for

Hence every may be uniquely written in the form:

.

That is is a basis of so .

Now I would argue that the *only *feature of this space is that . So for a finite set such as this one, with no additional structure at all, the algebra of functions can tell us everything about .

## Group Multiplication

Suppose we are told that is a finite set but that there is a binary relation such that satisfies the group axioms. We know from the last section that can recover the order of the group:

,

.

Can do more? Can recover the group law? Well if we make the assumption that the functions know what to do with for all then yes we can.

Choose to be an injective function; that is can distinguish between points:

.

Now evaluate each of the elements of to generate distinct complex numbers:

; .

Now evaluate the products:

; .

Now if

,

the injectivity of forces . Hence we can construct the multiplication tables.

In particular, if

; for all ,

then — the identity on .

## What about Infinite Cardinality

Let be a finite dimensional real vector space of dimension . Now the cardinality of is , but an algebra of functions (careful!) on can give us the dimension of .

Consider — the space of all linear maps on . Let be a basis for and suppose is a linear map on :

.

Let such that

.

Now

.

Hence is defined by what it does to the basis vectors. Say

,

so we need real numbers for each of the basis vectors to define a linear map. Therefore is of dimension (duh!). So if , then .

## Connectedness

Consider the interval . In the norm topology it is connected which means that we cannot represent as a union of disjoint subsets. Consider the continuous functions on , . Now call a *projection *if . That is

; for all ,

or .

Now this means that either takes the value or the value .

Suppose is a non-zero projection and set

.

Now . Now it should be clear that either or ; otherwise is not continuous as it will have jump discontinuities on the boundary of .

Hence the only projections on the connected set are the *trivial projections and . *

Now consider . Now is certainly disconnected but and are continuous non-trivial projections.

In general (!), if contains non-trivial projections, then is disconnected.

## Compactness

To consider compactness, equip a set with a metric and the induced topology. Suppose is a compact subset of such that is compact. For the purposes of this piece, we will suppose that compact just means that is closed and bounded ( will look like .)

Now let be the set of continuous functions on .

**Extreme Value Theorem**

*A continuous function on a compact set attains it’s absolute maximum and minimum on .*

In our example, every continuous function on attains its maximum on so in this case there exists a positive such that:

; for all (*).

Similarly, if is open or unbounded (i.e. not compact), then we can see that a condition such as (*) need not necessarily hold.

For example, is bounded but not closed. The continuous function is unbounded.

Also, is closed but not bounded. The identity function on , is unbounded.

So if we are given a space then we know that cannot be compact if contains unbounded functions.

## Hausdorffness

A space is *Hausdorff *if any two points there exists neighbourhoods of and , and say, such that and are disjoint.

With some extra assumptions on (namely that it is normal and locally compact), consider . If for all , there exists a pair such that then cannot be Hausdorff. Why?

**Urysohn’s Lemma**

*If are disjoint closed sets in a normal space , then there exists a continuous function such that .*

*Under the assumptions, Urysohn’s Lemma guarantees that there is a function such that and . That and exist are by local compactness.*

So we pick this function and say well , . Contradiction. That is must be in the same neighbourhood and Hausdorffness is destroyed.

## Non-Commutative Spaces

Note that in all these examples (except when we looked at ), the algebra of functions had a common structure:

**Vector Space**— for any complex valued function we can define point-wise the functions and for .**Normed Space**— there are various norms we could put on the algebra of functions; supremum norm, one norm, two norm, etc.**Inner Product Space**— again, for example, . We can go further if we pick the correct algebra of functions — we can pick the algebra of functions to be*complete,*so that we have a Hilbert Space (a complete inner product space).**Associative Algebra**— we can define pointwise a product on the algebra of functions.***-Algebra**— the algebra of functions takes on an involution , namely the conjugation:

.

Any algebra which has these five features (with a few conditions on how the norm interacts with the product and the involution), is known as a C*-algebra and by and large, the canonical algebra of functions on a space will have this structure.

Because of the commutativity of , the algebra of functions on is commutative:

; for all .

There is a beautiful theorem of Gelfand that states that every abelian C*-algebra is isomorphic to an algebra of functions on a space.

But what about *non-commutative C*-algebras?* How about:

*.*

Ordinarily we look at a space *, *and then look at the induced algebra of functions *. *Our work above has shown that often it is equally valid to look at a C*-algebra, say , and look at the induced space . Why not do this for a non-commutative C*-algebra ?

We could then say things about based on . We have to realise that is not going to look like our intuitive idea of space as a set of points. This is hard to imagine so don’t bother — looking at itself should be equally valid. For example, two definitions we could decide to use could be:

- is
*connected*if contains non-trivial projections - is
*compact*if contains unbounded elements

At this point I admit that this about the height of my understanding. I really should check this out properly, but I think Heisenberg’s uncertainty principle and a whole pile of modern quantum theory works on non-commutative spaces so in essence they are saying the very geometry of this universe is non-commutative!

For a proper and accurate account see

http://www.alainconnes.org/docs/book94bigpdf.pdf

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September 21, 2011 at 4:58 pm

An alternative quantisation of a Markov chain? or Why do we need Coalgebras? « J.P. McCarthy: Math Page[…] So essentially, this means that I thought you quantised objects, such as Markov chains, by replacing a commutative C*-algebra with a not-necessarily commutative one. (This roughly follows my interpretation as per this) […]