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

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 the identity of this multiplication.

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

Suppose the two linear functionals ''L''₁ and ''L''₂ are the same, say ''L.'' Then ''A_F'' becomes a commutative algebra with multiplication defined by
:$x \cdot y = L(x)y + L(y)x - L(x)L(y)e.$

Example

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

If ''f, g'' are in ''F''^''X'', ''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, ''F''^''X'' 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 ''F''^''X'' 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 ''F''^''X'' such that ''L''₁(''e'')= ''L''₂(''e'')= 1_''F''
For ''f, g'' in ''F''^''X'' define

:$f \cdot g = L_1(f)g + L_2(g)f - L_1(f) L_2(g) e = f(a)g + g(b)f - f(a)g(b)e.$

Then ''F''^''X'' becomes a non-commutative function algebra with the function ''e'' as the identity of multiplication.

Note that
:$(f \cdot g)(a) = f(a)g(a)\mbox (f \cdot g)(b) = f(b)g(b).$

FTA of Curves in the Complex Plane

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 $e(t) = 1, \forall \in [0, 1]$ we have for ''α,β'' in ''C''[0, 1],
:$\cdot = (0) + (1) - (0)(1)e$
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

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
:$f(t)=1 - t + it \mbox g(t)= \cos(2\pi t)+ i\sin(2\pi t)$

Since $g(0)=g(1)=1$ the circle ''g''
is a loop.
The line segment ''f'' starts from :$f(0)=1$
and ends at $f(1)= i$

Now, we get two ''f''-products
$f \cdot g \mbox g \cdot f$ given by

:$(f\cdot g)(t)=[-t+\cos (2\pi t)]+i[t+\sin(2\pi t)]$
and
:$(g\cdot f)(t)=[1-t - \sin (2\pi t)] +i[t-1+\cos(2\pi t)]$
See the Figure.

Observe that $f\cdot g \neq g\cdot f$ showing that
multiplication is non-commutative. Also both the products starts from $f(0)g(0)=1 \mbox f(1)g(1)= i.$

See also

* N-curve

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 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 201
Book of Beautiful Curves
* R. Sivaramakrishnan, ''Certain Number Theoretic Episodes in Algebra'', Chapman and Hall/CR
Certain Number Theoretic Episodes in Algebra
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Algebras