The convolution and class algebras 
Theorem 9.1. The convolution algebra
Proof: Since 
it is a Hilbert space as well as an algebra. For each 
Observe that 
Thus, 
We claim that 
hence
From this we can conclude that the image of 
-QED
The proof of Theorem 8.1 is the linear version of Cayley’s theorem from group theory: instead of representing 
Corollary 9.2. The class algebra
Proof: Since
-QED
For an example illustrating how the proof of Theorem 8.1 works, let us take 
be its expansion in the group basis, so that
Problem 9.1. Show by direct calculation that the matrix of 
In the case where 
To see begin to see where a projection basis of 
Theorem 9.3. A linear transformation 
is a group homomorphism taking values in 
Problem 9.2. Prove Theorem 8.3.
Definition 9.1. A group homomorphism 
Observe first of all that 
Problem 9.3. Determine the Fourier dual of the symmetric group
For abelian groups the situation is much better and in fact we always have 
where 
That is,
where 
Indeed, the parameterization 
Theorem 9.4. For every 
where 
Proof: For any 
so 
-QED
We now have a special subset 
Theorem 9.5. The set 
Proof: For any 
where
Thus if 
and if 
where the denominator of each fraction is nonzero and the numerator is zero, because 
-QED
The orthogonal basis 
![X=[\chi^\lambda_\alpha]](https://s0.wp.com/latex.php?latex=X%3D%5B%5Cchi%5E%5Clambda_%5Calpha%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
is an orthonormal basis of the convolution algebra
is even better, for the following reason.
Theorem 9.6. The elements of the basis 
Proof: For any
For any
The internal sum is
where the final equality is Theorem 8.3. Thus
-QED
Since we know that any algebra with a basis of orthogonal projections is isomorphic to the function algebra (Lecture 3), Theorem 8.4 gives the promised second proof of the fact that the convolution algebra
, every partition of 
is called a class function if it is constant on conjugacy classes, meaning that 
for all 
for all 
is a class function on 
if and only if it is constant on conjugacy classes.
For any 
, we have = \sum\limits_{h \in G} A(gh^{-1})B(h) = \sum\limits_{h \in G} A(g(gh)^{-1}) B(gh) = \sum\limits_{h \in G} A(gh^{-1}g^{-1})B(gh),](https://s0.wp.com/latex.php?latex=%5BAB%5D%28g%29+%3D+%5Csum%5Climits_%7Bh+%5Cin+G%7D+A%28gh%5E%7B-1%7D%29B%28h%29+%3D+%5Csum%5Climits_%7Bh+%5Cin+G%7D+A%28g%28gh%29%5E%7B-1%7D%29+B%28gh%29+%3D+%5Csum%5Climits_%7Bh+%5Cin+G%7D+A%28gh%5E%7B-1%7Dg%5E%7B-1%7D%29B%28gh%29%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
simply permutes the terms of the sum. Continuing the calculation, we have = \sum\limits_{h \in G} A(gh^{-1}g^{-1})B(gh) =\sum\limits_{h \in G} A(h^{-1})B(gh) = \sum\limits_{h \in G} B(gh)A(h^{-1}),](https://s0.wp.com/latex.php?latex=%5BAB%5D%28g%29+%3D+%5Csum%5Climits_%7Bh+%5Cin+G%7D+A%28gh%5E%7B-1%7Dg%5E%7B-1%7D%29B%28gh%29+%3D%5Csum%5Climits_%7Bh+%5Cin+G%7D+A%28h%5E%7B-1%7D%29B%28gh%29+%3D+%5Csum%5Climits_%7Bh+%5Cin+G%7D+B%28gh%29A%28h%5E%7B-1%7D%29%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
 = \sum\limits_{h \in G} B(gh^{-1})A(h) = [BA](g),](https://s0.wp.com/latex.php?latex=%5BAB%5D%28g%29+%3D+%5Csum%5Climits_%7Bh+%5Cin+G%7D+B%28gh%5E%7B-1%7D%29A%28h%29+%3D+%5BBA%5D%28g%29%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
is a central function; we will prove it is constant on conjugacy classes. Since 
commutes with all functions in 
it commutes with every elementary function 
. Since = \sum\limits_{k \in G}Z(hk^{-1})E_g(k)=Z(hg^{-1}),](https://s0.wp.com/latex.php?latex=%5BZE_g%5D%28h%29+%3D+%5Csum%5Climits_%7Bk+%5Cin+G%7DZ%28hk%5E%7B-1%7D%29E_g%28k%29%3DZ%28hg%5E%7B-1%7D%29%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
 = \sum\limits_{k\in G}E_g(hk^{-1})Z(k)=Z(g^{-1}h),](https://s0.wp.com/latex.php?latex=%5BE_gZ%5D%28h%29+%3D+%5Csum%5Climits_%7Bk%5Cin+G%7DE_g%28hk%5E%7B-1%7D%29Z%28k%29%3DZ%28g%5E%7B-1%7Dh%29%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
for all 
. Thus 
be a set parameterizing the conjugacy classes of 
let 
be the indicator function of 
so 
span the class algebra of 
constant on conjugacy classes can be written
denotes the value 
for any 
We will show that 
is a partition of 
is an orthogonal (but not orthonormal) set of functions in ![[c_{\alpha\beta\gamma}]](https://s0.wp.com/latex.php?latex=%5Bc_%7B%5Calpha%5Cbeta%5Cgamma%7D%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
of 
i.e. 
For any 
and any 
, show that
counts solutions to the equation 
in 
and 
are required to belong to 
and 
respectively.
be an algebra.
be a finite nonempty set.
 = A(x)B(x), \quad x \in X,](https://s0.wp.com/latex.php?latex=%5BAB%5D%28x%29+%3D+A%28x%29B%28x%29%2C+%5Cquad+x+%5Cin+X%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
For each 
define the corresponding elementary function 
by
where 
is the =E_x(z)E_y(z)=\delta_{xz}\delta_{yz}=\delta_{xy}E_x(z).](https://s0.wp.com/latex.php?latex=%5BE_xE_y%5D%28z%29%3DE_x%28z%29E_y%28z%29%3D%5Cdelta_%7Bxz%7D%5Cdelta_%7Byz%7D%3D%5Cdelta_%7Bxy%7DE_x%28z%29.&bg=FFFFFF&fg=000&s=0&c=20201002)
is a linearly independent set in 
can be written as
are said to be isomorphic if there exists a 
are isomorphic.
be the function algebra of this set, and define a linear transformation
, where 
is the elementary function corresponding to 
is also an algebra homomorphism. Indeed, by Theorem 1.2 and linearity of 
Next,
then the linear transformation
rather than 
Thus,
so defined is an algebra isomorphism. Note that the Fourier transform on 
, its expansion in the Fourier basis is written
We thus have
The function 
is called the Fourier transform of the algebra element 
The practical value of the Fourier transform lies in the fact that 
is the product of 
, which might be convoluted and difficult to calculate. On the other hand, the pointwise product of the corresponding functions 
is very simple, both conceptually and computationally. 
, the complex number system, elements of which are called scalars and denoted by lower-case Greek letters,
and 
as usual, and we write 
for the 
whose values are denoted by concatenating its arguments: 
. Associativity means that the symbol 
is unambiguous because its two possible meanings coincide:
denote the zero vector in 
for all 
such that
is distinct from the additive unit 
In fact, there is only one multiplicative unit. 
be multiplicative units in 
.
for the unique multiplicative unit. Later on, we may omit the subscript and simply write 
such that 
.
are such that 
. Prove that 
.
is the inverse of 
This is written 
and 
.
range over ![[\gamma_{xyz}]](https://s0.wp.com/latex.php?latex=%5B%5Cgamma_%7Bxyz%7D%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
of complex numbers called the multiplication tensor of 
The elements of this three-tensor are called the connection coefficients of 
This set of 
numbers completely determines multiplication in 
whose values are denoted by a superscript asterisk: 
. Antilinearity means that conjugation interacts with the vector space operations according to the rule
![[\eta_{xy}]](https://s0.wp.com/latex.php?latex=%5B%5Ceta_%7Bxy%7D%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
relative to the the basis 
of invertible elements in an algebra 
is invertible, and in fact 
.
.
.
.
.
of all selfadjoint elements in an algebra 
with 
selfadjoint. We say that 
is called a projection. Projections 
are said to be orthogonal if 
are such that 
where 
we have
we must have 
for each 
of all unitary elements in 
be its decomposition into real and imaginary parts. Then 
be any two selfadjoint elements, and set 
commute even if they are not selfadjoint. Then we can write 
and 
where 
are selfadjoint and thus commute with one another. Thus, 
is said to be an algebra homomorphism if
are isomorphic if there is 
to indicate that 
together with an associative, bilinear, unital multiplication and a linear, involutive conjugation.
We write 
as 
and 
we declare
for some 
and 
be real elements of 
is also a real element of 
and 
are selfadjoint elements of 
we have that the real part of 
and its imaginary part is 
Since 
commute, 
of 
is the smallest subalgebra of 
are called central elements.
be any two central elements and 
, we have
.
(maximally commutative, minimally noncommutative). Any algebra 
have the lowest commutativity index; such maximally noncommutative algebras are called 
is the set of all elements in 
.
is indeed a subalgebra of 
are subalgebras of 
then 
and whose “largest” element is 
, a relation on 
of 
to denote the membership of 
in 
For example, you are familiar with the notion of an 
with the following properties :
for all 
;
iff 
;
then 
.
for all 
and 
then 
;
and 
then 
.
consisting of a set together with a partial order on it is called a 
by definition means 
. Importantly, note the use of the adjective “partial” – Definition 2.3 allows for the possibility that there are pairs of elements 
for which neither 
by inclusion:
, then 
subsets of 
is the largest subset of 
is the smallest subset of 
set is the only subset of 
. Similarly, the whole set 
is the only set satisfying 
to be the set of all vector subspaces of 
is an 
as intersection, because the intersection of two subspaces is again a subspace. However, the union of two subspaces is not, and we have to modify the definition to 
, as the empty subset of 
to be the set of all subalgebras of 
is an induced subposet of 
of subalgebras of 
into a lattice, because the span of the union 
of two subalgebras of 
, define 
to be the intersection of all subalgebras of 
be the set of all commutative subalgebras of 
This definition will fail if one of 
is not contained in any larger commutative subalgebra of 
with the following property: if 
is a commutative subalgebra with 
, then 
.
. If 
such that 
In this case, the algebra generated by 
is a commutative subalgebra of 
and this contradicts the maximality of 
Then for a commutative subalgebra 
we have
(note that the conclusion of Problem 2.1 was used here) .
. Annals of Mathematics 18 (1916), p. 70-73.