Functional-theoretic algebra: Difference between revisions
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Let ''A<sub>F</sub>'' be a [[vector space]] over a [[field (mathematics)|field]] ''F'', and let ''L''<sub>1</sub> and ''L''<sub>2</sub> be two [[linear functional]]s on A<sub>F</sub> with the property ''L''<sub>1</sub>(''e'') = ''L''<sub>2</sub>(''e'') = 1<sub>''F''</sub> for some ''e'' in ''A<sub>F</sub>''. We define multiplication of two elements ''x'', ''y'' in ''A<sub>F</sub>'' by |
Let ''A<sub>F</sub>'' be a [[vector space]] over a [[field (mathematics)|field]] ''F'', and let ''L''<sub>1</sub> and ''L''<sub>2</sub> be two [[linear functional]]s on A<sub>F</sub> with the property ''L''<sub>1</sub>(''e'') = ''L''<sub>2</sub>(''e'') = 1<sub>''F''</sub> for some ''e'' in ''A<sub>F</sub>''. We define multiplication of two elements ''x'', ''y'' in ''A<sub>F</sub>'' by |
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:<math> x \cdot y = L_1(x)y + L_2(y)x - L_1(x) L_2(y) e. </math> |
:<math> x \cdot y = L_1(x)y + L_2(y)x - L_1(x) L_2(y) e. </math> |
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It can be verified that the above multiplication is associative and that ''e'' is a unit element for this multiplication. So, A<sub>F</sub> forms an associative algebra with unit ''e'' and is called a ''functional-theoretic algebra''. |
It can be verified that the above multiplication is associative and that ''e'' is a unit element for this multiplication. So, A<sub>F</sub> forms an associative algebra with unit ''e'' and is called a ''functional-theoretic algebra''. The product ''x\cdot y'' is called the ''f-product''. |
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==Example== |
==Example== |
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''X'' is a nonempty set and ''F'' a field. ''A''<sub>''F''</sub> is the set of functions from ''X'' to ''F''. |
''X'' is a nonempty set and ''F'' a field. ''A''<sub>''F''</sub> is the set of functions from ''X'' to ''F''. |
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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 |
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:<math>f\cdot g</math> as above. |
:<math>f\cdot g</math> as above. |
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A curve ''f'' is a ''loop'' at ''z'' if |
A curve ''f'' is a ''loop'' at ''z'' if <math> f(0) = f(1)=z </math> |
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Let us take three loops at ''z = 1'' and find their f-products. |
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1. The Unit Circle :<math> u: x^2 + y^2 = 1 </math> |
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2. The Rhodonea Curve :<math> \lambda: r = \cos(2\theta)</math> |
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3. The Astroid:<math> \alpha: x^{\frac{2}{3}} +y^{\frac{2}{3}}=1</math> |
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Revision as of 05:39, 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. The product x\cdot y is called the f-product.
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 Let us take three loops at z = 1 and find their f-products. 1. The Unit Circle : 2. The Rhodonea Curve : 3. The Astroid:
References
- Sebastian Vattamattam and R. Sivaramakrishnana, "A Note on Convolution Algebras", in Recent Trends in Mathematical Analysis, Allied Publishers, 2003.