Functional-theoretic algebra

Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals. In general, it is a non-commutative algebra. It becomes commutative when the two functionals are the same.

Definition

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

 

It can be verified that the above multiplication is associative and that e is the identity of this multiplication.

So, AF forms an associative algebra with unit e and is called a functional theoretic algebra(FTA).

Suppose the two linear functionals L1 and L2 are the same, say L. Then AF becomes a commutative algebra with multiplication defined by

 

Example

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X is a nonempty set and F a field. FX is the set of functions from X to F.

If f, g are in FX, x in X and α in F, then define

 

and

 

With addition and scalar multiplication defined as this, FX 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) = 1F for all x in X.

Define L1 and L2 from FX to F by L1(f) = f(a) and L2(f) = f(b).

Then L1 and L2 are two linear functionals on FX such that L1(e)= L2(e)= 1F For f, g in FX define

 

Then FX becomes a non-commutative function algebra with the function e as the identity of multiplication.

Note that

 

FTA of Curves in the Complex Plane

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Let C denote the field of Complex numbers. A continuous function γ from the closed interval [0, 1] of real numbers to the field C is called a curve. The complex numbers γ(0) and γ(1) are, respectively, the initial and terminal points of the curve. If they coincide, the curve is called a loop. The set V[0, 1] of all the curves is a vector space over C.

We can make this vector space of curves into an algebra by defining multiplication as above. Choosing   we have for α,β in C[0, 1],

 

Then, V[0, 1] is a non-commutative algebra with e as the unity.

We illustrate this with an example.

Example of f-Product of Curves

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Let us take (1) the line segment joining the points (1, 0) and (0, 1) and (2) the unit circle with center at the origin. As curves in V[0, 1], their equations can be obtained as

 

Since   the circle g is a loop. The line segment f starts from :  and ends at  

Now, we get two f-products   given by

 

and

 

See the Figure.  

Observe that   showing that multiplication is non-commutative. Also both the products starts from  

See also

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References

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  • Sebastian Vattamattam and R. Sivaramakrishnan, A Note on Convolution Algebras, in Recent Trends in Mathematical Analysis, Allied Publishers, 2003.
  • Sebastian Vattamattam and R. Sivaramakrishnan, Associative Algebras via Linear Functionals, Proceedings of the Annual Conference of K.M.A., Jan. 17 - 19, 2000, pp. 81-89
  • Sebastian Vattamattam, Non-Commutative Function Algebras, in Bulletin of Kerala Mathematical Association, Vol. 4, No. 2, December 2007
  • Sebastian Vattamattam, Transforming Curves by n-Curving, in Bulletin of Kerala Mathematics Association, Vol. 5, No. 1, December 2008
  • Sebastian Vattamattam, Book of Beautiful Curves, January 2015

Book of Beautiful Curves

  • R. Sivaramakrishnan, Certain Number Theoretic Episodes in Algebra, Chapman and Hall/CRC

Certain Number Theoretic Episodes in Algebra