Functional-theoretic algebra: Difference between revisions
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:<math> 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). </math> |
:<math> 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). </math> |
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We use the above ideas to construct a non-commutative algebra of curves in the complex plane |
We use the above ideas to construct a non-commutative algebra of curves in the complex plane ''C''. |
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A curve is a continuous function from the closed interval [0, 1] to ``C’’. |
A curve is a continuous function from the closed interval [0, 1] to ``C’’. |
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The set ``C[0, 1]’’ of curves in ``C’’ is a vector space over ``C’’. It becomes a non-commutative algebra by defining |
The set ``C[0, 1]’’ of curves in ``C’’ is a vector space over ``C’’. It becomes a non-commutative algebra by defining |
Revision as of 05:32, 23 January 2008
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
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.
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
and
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
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
- as above.
A curve ``f’’ is a ``loop’’ at ``z’’ if :
References
- Sebastian Vattamattam and R. Sivaramakrishnana, "A Note on Convolution Algebras", in Recent Trends in Mathematical Analysis, Allied Publishers, 2003.