Functional-theoretic algebra

In mathematics, a functional-theoretic algebra is a unital associative algebra whose multiplication is defined by the action of two linear functionals.

Let A_F be a vector space over a field F, and let L₁ and L₂ be two linear functionals on A_F with the property L₁(e) = L₂(e) = 1_F for some e in A_F. We define multiplication of two elements x, y in A_F by

$x\cdot y=L_{1}(x)y+L_{2}(y)x-L_{1}(x)L_{2}(y)e.$

It can be verified that the above multiplication is associative and that e is a unit element for this multiplication. So, A_F forms an associative algebra with unit e and is called a functional-theoretic algebra. The product :$x\cdot y$ is called the f-product.

Example

X is a nonempty set and F a field. A_F is the set of functions from X to F. If f, g are in A_F, x in X and α in F, then define

$(f+g)(x)=f(x)+g(x)\,$

and

$(\alpha f)(x)=\alpha f(x).\,$

With addition and scalar multiplication defined as this, A_F is a vector space over F. Now, fix two elements a, b in X and define a function e from X to F by e(x) = 1_F for all x in X. Define L₁ and L₂ from A_F to F by L₁(f) = f(a) and L₂(f) = f(b). Then L₁ and L₂ are two linear functionals on A_F such that L₁(e)= L₂(e)= 1_F For f, g in A_F define

$f\cdot g=L_{1}(f)g+L_{2}(g)f-L_{1}(f)L_{2}(g)e=f(a)g+g(b)f-f(a)g(b).$

We use the above ideas to construct a non-commutative algebra of curves in the complex plane C. A curve is a continuous function from the closed interval [0, 1] to C. The set C[0, 1] of curves in C is a vector space over C. It becomes a non-commutative algebra by defining

$f\cdot g$ as above.

A curve f is a loop at z if $f(0)=f(1)=z$ Let us take three loops at z = 1 and find their f-products.

1. The Unit Circle :$u:x^{2}+y^{2}=1$

2. The Rhodonea Curve :$\lambda :r=\cos(2\theta )$

3. The Astroid:$\alpha :x^{\frac {2}{3}}+y^{\frac {2}{3}}=1$

References

• Sebastian Vattamattam and R. Sivaramakrishnana, "A Note on Convolution Algebras", in Recent Trends in Mathematical Analysis, Allied Publishers, 2003.