Functional-theoretic algebra

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

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, AF forms an associative algebra with unit e and is called a functional-theoretic algebra.

[edit] Example

X is a nonempty set and F a field. AF is the set of functions from X to F. If f, g are in AF, 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, AF 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 AF to F by L1(f) = f(a) and L2(f) = f(b). Then L1 and L2 are two linear functionals on AF such that L1(e)= L2(e)= 1F For f, g in AF 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).

Curves in the Complex Plane Let C denote the field of complex numbers. A continuous function f from the closed interval [0, 1] of real numbers to the field C is called a curve. The complex numbers f(0) and f(1) are, respectively, the initial and terminal points of the curve. If they coincide, the curve is called a loop. The range of f describes an unbroken path in the complex plane, which sometimes is called the trace of f. The set of all the curves, denoted by C[0, 1], is a vector space over C.

We can make this vector space of curves into a non-commutative algebra by defining multiplication as above. Choosing e(t) = 1 for all t in [0, 1] we have for f, g in C[0, 1], f \cdot g = f(0)g + g(1)f - f(0)g(1) We illustrate this with an example.

Example Let us take (1) the unit circle with center at the origin and radius one, and (2) the the line segment joining the points (1, 0) and (0, 1). As curves in C, their equations can be obtained as :u(t) = cos(2πt) + isin(2πt) and f(t) = 1 − t + it Since u(0) = u(1) = 1 the circle u is a loop. The line segment f starts from f(0) = 1 and ends at f(1) = i

Now, we get two f-products u\middot f and f\middot u given by

(u\cdot f)(t)=[1-t - \sin (2\pi t)] +i[t-1+\cos(2\pi t)] and
(f\cdot u)(t)=[-t+\cos (2\pi t)]+i[t+\sin(2\pi t)]

For the traces of the two curves and their products, see the figure.

Image:F-products copy.jpg


Observe that u\cdot f \neq f\cdot u showing thet multiplication is noncommutative.


[edit] References

  • 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 Mathematics Association, Vol. 4, No. 2, December 2007


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