|
|
||||
|
7 ALGEBRAIC PROPERTIES OF CONVOLUTION 277
(14.7.3) If the sequence (u^..., un) is convolvable, then so is the sequence |
||||
|
|
||||
|
(14.7.3.1) (//! * fi2 * • • • * uny = £in * #„_! * • • • * /V
For (14.1.4 and 13.7.10) we have
|
||||
|
|
||||
|
|
f-r
|
|
||
|
|
||||
|
if and only if
|
||||
|
|
||||
|
j-r
|
||||
|
|
||||
|
and these two integrals are equal.
On the other hand, if the sequence (A, u) is convolvable, it does not
necessarily follow that (/*, A) is convolvable (Problem 2). But if G is com- mutative this will be the case, and we shall have A * \JL = fj. * A.
In particular, it follows from the preceding results that
(14.7.4) On the set M£(G) of bounded measures on G, the law of composition
(A, p) i-> A * IJL (together with the vector space structure) defines a C-algebra structure\ the unit element is the Dirac measure se at the neutral element e ofG. The set M£(G) of compactly supported measures on G is a subalgebra of Mc(G). The algebra M£(G) is commutative if and only ifG is commutative.
The fact that G is commutative if M£(G) is commutative follows from the
formula (14.6.1.2).
If G is discrete, the algebra M£(G) consists of all linear combinations
]T as8&, where as = 0 for all but a finite number of points s e G (3.16.3), and
seG
the formula (14.6.1.2) shows that
|
||||
|
|
||||
|
seG / \seG
|
||||
|
|
||||
|
This is what is called in algebra the group algebra of the group G over the
field C. |
||||
|
|
||||