The convolution algebra 
The degree of non commutativity of
The orbits of this action are the conjugacy classes of
Problem 8.1. Prove that
For the function algebra 
Definition 8.2. A function 
The following gives an alternative way to think about class functions as functions which are insensitive to any noncommutativity present in
Theorem 8.1. A function 
Proof: Suppose first that 
Conversely, suppose
-QED
We can now characterize the center of
Theorem 8.2. A function 
Proof: Suppose first that 
 = \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)
where the second inequality follows from the fact that the substitution 
 = \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)
where the second inequality follows from the fact that
 = \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)
as required.
Now suppose that 
 = \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)
and
 = \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)
the centrality of 
-QED
Thus, the set of all functions constant on conjugacy classes of 
Just as the convolution algebra 
and this is a subset of the power set of 
Equivalently, in terms of the group basis 
Then, the functions 
where 
It is clear that
Since 
which shows that 
In conclusion, if
Problem 8.3. Consider the multiplication tensor ![[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)
Thus, the multiplication tensor of 
be an algebra.
be a finite nonempty set.
-valued functions on  = 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)
are defined 
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 a vector space basis of 
be the function algebra of this set, and define a linear transformation
, where 
is the elementary function corresponding to 
. This is a vector space isomorphism, since it maps a basis of 
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) .