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 "L"₁ and "L"₂ be two linear functionals on A_F with the property "L"₁("e") = "L"₂("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".

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 "L"₁ and "L₂" from "A"_"F" to "F" by "L"₁("f") = "f"("a") and "L"₂("f") = "f"("b"). Then "L"₁ and "L"₂ are two linear functionals on "A"_"F" such that "L"₁("e")= "L"₂("e")= 1_"F"For "f, g" in "A"_"F" define

:$f\; cdot\; g\; =\; L\_1(f)g\; +\; L\_2(g)f\; -\; L\_1(f)\; L\_2(g)\; e\; =\; f(a)g\; +\; g(b)f\; -\; f(a)g(b).$

Curves in the Complex PlaneLet "C" denote the field of
complex numbers. A continuous function "f" from the closedinterval [0, 1] of real numbers to the field "C" is called acurve. The complex numbers "f"(0) and "f"(1) are, respectively,the initial and terminal points of the curve. If they coincide, thecurve is called a loop. The range of "f" describes an unbrokenpath in the complex plane, which sometimes is called the "trace"of "f." The set of all the curves, denoted by "C" [0, 1] , is avector space over "C".

We can make this vector space of curves into a non-commutativealgebra 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 illustratethis with an example.

ExampleLet us take (1) the unit circle with center at theorigin 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 :$u(t)=cos(2pi\; t)+isin(2pi\; t)$ 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 :$(ucdot\; f)(t)=\; [1-t\; -\; sin\; (2pi\; t)]\; +i\; [t-1+cos(2pi\; t)]$ and :$(fcdot\; u)(t)=\; [-t+cos\; (2pi\; t)]\; +i\; [t+sin(2pi\; t)]$

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

Observe that $ucdot\; f\; eq\; fcdot\; u$ showing thet multiplication is noncommutative.

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