[This is the first of probably very many math related posts I’m going to make. I intend on posting notes for lectures and talks that I find either particularly interesting to me. Hopefully, they will be of interest to others as well. Some details are missing from this post and the output is pretty ugly. I’ll correct these when I stop being lazy. Thanks go out to Eusebio for giving an awesome talk.]

This post are the lecture notes from a seminar talk given by Eusebio Gardella of the same title, during the Analysis seminar at the University of Oregon on May 10, 2011.

Let be an infinite dimensional, separable Hilbert space. We denote the set of bounded linear operators of as . has a two-sided closed ideal consisting of the compact operators of (i.e. operators that map bounded sets into precompact sets). is the closure of the finite rank operators, which forms a (not closed) two-sided ideal of . We start off with a well-known theorem that doubles as a definition.

**Atkinson’s Theorem**

For any bounded linear operator of , the following are equivalent:

- There exists a bounded linear operator of such that and are compact operators.
- The image of in the quotient map is invertible.
- , are finite dimensional and is closed.

It should be immediately clear that (1) and (2) are equivalent. If satisfies any of the conditions above, we say that is a Fredholm operator and the Fredholm index of is defined by

If we take and define the unilateral shift by , where is the sequence with 1 in the th coordinate and else. is injective, and . So is a Fredholm operator and .

We denote as the set of Fredholm operators of . Note that forms a semi-group under composition (but is not additively closed). We have the following properties of the index map:

- is locally constant.
- .
- for all compact operators .
- If , then is invertible for some compact operator .
- .

This last statement holds because .

**Theorem 2**

The index characterizes the connected components of , i.e. if and only if and are homotopic (i.e. in the same connected component).

This follows from the properties above and the fact that is connected. This in turn follows from the fact that , which are both connected.

So we now turn to the commutative version of the theorem that we will generalize.

**Atiyah-Janich Theorem**

Let be a compact, Hausdorff space. We denote as the set of homotopy classes of continuous maps from into . There is a a group isomorphism

Note that if is just a point, then and the index map is the Fredholm index. Also note that there is a natural isomorphism .

**Non-commutative Atiyah-Janich Theorem**

Let be a -unital -algebra. There is a group isomorphism

So what does all this mean? is the Grothendieck group of , which is the set of isomorphism classes of finitely generated projective modules over with direct sum being the semi-group operation. We can think about as the set of formal differences of elements in .

Note that in the case where , finitely generated projective modules over are the same as finite dimensional vector spaces. The isomorphism classes of vector spaces correspond to their dimension and the direct sum corresponds to addition of dimensions. So and so .

To describe requires some definitions. A right Hilbert -module over is a triple consisting of a Banach space , a continuous right action and an inner product such that some conditions are satisfied.

An example is if is an ideal of , then is a -module by the action being right multiplication and .

Another example is . and . This is known as the standard Hilbert -module.

A map between Hilbert modules is said to be adjointable if there exists such that . We denote the set of adjointable maps, we have a norm, which is reminiscent of the operator norm . Further, is a -algebra.

Given Hilbert modules, for each and , we denote by the map . This should be reminiscent of rank one operators. is adjointable with . We denote . is an ideal of , despite in general, elements of not being compact operators. To distinguish between the two notions, we say that elements of are -compact.

Now we say that is Fredholm, if there is a such that and are -compact.

**Atkinson-Mingo Theorem**

For , the following are equivalent:

- is Fredholm,
- There exists such that has closed range and , are finitely generated projective -modules over .

Finally, we define .