A common theme in mathematics is to replace the study of an object with the study of some category that can be built from that object. For example, we can
- replace the study of a group
with the study of its category
of linear representations,
- replace the study of a ring
with the study of its category
of
- replace the study of a topological space
with the study of its category
of sheaves,
and so forth. A general question to ask about this setup is whether or to what extent we can recover the original object from the category. For example, if
Today we will prove an even simpler reconstruction theorem.
Theorem: A group 
![k[G]](http://s0.wp.com/latex.php?latex=k%5BG%5D&bg=ffffff&fg=333333&s=0 "k[G]")
, which is the free 
-module on 
of functions 
with pointwise multiplication. 
and when ![\mathbb{C}[G]](http://s0.wp.com/latex.php?latex=%5Cmathbb%7BC%7D%5BG%5D&bg=ffffff&fg=333333&s=0 "\mathbb{C}[G]")
, the corresponding state is a noncommutative refinement of 
, the corresponding state is by definition integration with respect to normalized Haar measure on 
it is possible to at least describe integration against Haar measure for a dense subalgebra of the algebra of class functions on 
of (finite-dimensional continuous unitary) representations of 
being analogous to the inner product); what it actually behaves like is a random algebra, namely the random algebra of class functions on 
of a group 
for all 
.
of surjective group homomorphisms 
. In other words, they are precisely the subgroups you can quotient by and get another group. This strikes me as backwards. The motivation to construct quotient groups should come first.
,
,
?
, where 
is an arbitrary set, shows that there exist groups of arbitrarily large cardinality, so no particular group can be universal. 
(which you’ll recall we assumed way back in 
. 
. Part of the reason it is so simple to understand is that 
of its elements, showing that it can be understood either in terms of the group of elements of norm 
in 
(that is, the unitary group 
) or the imaginary subspace of 
-sphere 
. It turns out that there is a picture completely analogous to the picture above, but with 
: that is, 
), and there is an exponential map from the imaginary subspace of 
lets us handle elements of 
almost as easily as we handle elements of 
. 
