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In the mathematical theory of [[free probability]], the notion of '''free independence''' was introduced by [[Dan Voiculescu (mathematician)|Dan Voiculescu]].<ref>D. Voiculescu, K. Dykema, A. Nica, "Free Random Variables", CIRM Monograph Series, AMS, Providence, RI, 1992</ref> The definition of free independence is parallel to the classical definition of [[independence (probability)|independence]], except that the role of Cartesian products of [[measure space]]s (corresponding to [[tensor product]]s of their function algebras) is played by the notion of a [[free product]] of (non-commutative) probability spaces.
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In the context of Voiculescu's free probability theory, many classical-probability theorems or phenomena have free probability analogs: the same theorem or phenomenon holds (perhaps with slight modifications) if the classical notion of independence is replaced by free independence.  Examples of this include: the free central limit theorem; notions of [[free convolution]]; existence of [[free stochastic calculus]] and so on.
 
Let <math>(A,\phi)</math> be a [[non-commutative probability space]], i.e. a [[Identity element|unital]] [[Algebra over a field|algebra]] <math>A</math> over <math>\mathbb{C}</math> equipped with a [[Unital map|unital]] [[linear functional]] <math>\phi:A\to\mathbb{C}</math>.  As an example, one could take, for a probability measure <math>\mu</math>,
 
: <math>A = L^\infty(\mathbb{R},\mu),\phi(f) = \int f(t)\,d\mu(t).</math>
 
Another example may be <math>A=M_N</math>, the algebra of <math>N\times N</math> matrices with the functional given by the normalized trace <math>\phi=\frac{1}{N}Tr</math>. Even more generally, <math>A</math> could be a [[von Neumann algebra]] and <math>\phi</math> a state on <math>A</math>. A final example is the [[group algebra]] <math>A=\mathbb{C}\Gamma</math> of a (discrete) [[Group (mathematics)|group]] <math>\Gamma</math> with the functional <math>\phi</math> given by the group trace <math>\phi (g) = \delta_{g=e},g\in \Gamma</math>.
 
Let <math>\{A_i : i\in I\}</math>  be a family of unital subalgebras of <math>A</math>.
 
'''Definition'''. The family <math>\{A_i : i\in I\}</math>  is called ''freely independent'' if
<math>\phi(x_1 x_2 \cdots x_n) =0 </math>
whenever <math>\phi(x_j)=0</math>, <math>x_j \in A_{i(j)}</math> and <math>i(1)\neq i(2), i(2)\neq i(3),\dots</math>.
 
If <math>X_i\in A</math>, <math>i\in I</math> is a family of elements of <math>A</math> (these can be thought of as random variables in <math>A</math>), they are called
 
''freely independent''  if the algebras <math>A_i</math> generated by <math>1</math> and <math>X_i</math> are freely independent.
 
== Examples of free independence ==
 
* Let <math>\Gamma</math> be the [[free product]] of groups <math>\Gamma_i,i\in I</math>, let <math>A=\mathbb{C}\Gamma</math> be the group algebra, <math>\phi(g)=\delta_{g=e}</math> be the group trace, and set <math>A_i=\mathbb{C}\Gamma_i\subset A</math>.  Then <math>A_i:i\in I</math> are freely independent.
 
* Let <math>U_i(N),i=1,2</math> be <math>N\times N</math> unitary [[random matrices]], taken independently at random from the <math>N\times N</math> [[unitary group]] (with respect to the [[Haar measure]]).  Then <math>U_1(N),U_2(N)</math> become asymptotically freely independent as <math>N\to\infty</math>. (Asymptotic freeness means that the definition of freeness holds in the limit as <math>N\to\infty</math>).
 
* More generally, independent [[random matrices]] tend to be asymptotically freely independent, under certain conditions.
 
== References ==
 
{{Reflist}}
 
[[Category:Functional analysis]]
[[Category:Free probability theory]]
[[Category:Free algebraic structures]]

Latest revision as of 17:12, 29 March 2014

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