Notation for a multiplicative algebra
I am attempting to identify what the proper symbols for a multiplicative algebra need to be. I've seen a number of unusual and also contradictory means for creating these points in the literary works, yet they are all either also specialized or generate added luggage.
To be extra details, offered a ring R and also an R-module, M, with a limited basis B, we can recognize the components of M with features of the kind $f, g : B \to R$. After that we construct the multiplicative algebra on M according to the adhering to regulation :
$$ (f \, g)(x) = f(x) g(x) $$
My inquiry is straightforward : What is the typical symbols for this building and construction? I've seen a number of unusual kinds. As an example, in case where R is the area of intricate numbers, we can write something like [; C ^ k (B ); ] for this algebra. Or if we are managing matrices, we can speak about Hadamard items.
Yet can we write something less complex? I would actually similar to to do something along the lines of say; specify R (B ) to be this multiplicative algebra and also be performed with it. The factor for this is that if we are collaborating with something like Pontryagin duality or depiction concept, the set B might be a group/monoid/whatever, and also there can be numerous algebras we specified over it.
To highlight why this is a concern, allow $B = \mathbb R$ be the team of reals under enhancement, after that we can specify both a convolution algebra, $R[ \mathbb R ]$ and also a multiplicative algebra $ R( \mathbb R )$ which are understood the very same R-module, yet have entirely various frameworks. (Though they are connected by Pontryagin duality in case where $R = \mathbb C$. )