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  "body": "\n## Definition\n\nA *noncommutative probability space* is a pair $(A, \\varphi)$ where $A$ is a unital algebra over $\\C$ and $\\varphi \\colon A \\to \\C$ is a linear functional with $\\varphi(1) = 1$.\nElements of $A$ are called noncommutative random variables.\nLet $(A_i)_{i \\in I}$ be a family of unital subalgebras of $A$.\nThey are *freely independent*, or *free*, if the following holds.\nWhenever $n \\ge 1$, $a_j \\in A_{i(j)}$ and $\\varphi(a_j) = 0$ for all $j$, and consecutive indices differ, $i(1) \\ne i(2) \\ne \\cdots \\ne i(n)$, then\n$$\\varphi(a_1 a_2 \\cdots a_n) = 0.$$\nElements of $A$ are free if the unital subalgebras they generate are free.\n*Free probability* is the study of noncommutative probability spaces and of free independence.\n\n## Remarks\n\nExample: let $F_2$ be the [[free group|0KTW26C80J]] on generators $a$ and $b$, let $A = \\C[F_2]$ be its group algebra, and let $\\varphi(\\sum_g c_g g) = c_e$, the coefficient of the identity.\nThen the subalgebras $A_1 = \\C[\\langle a \\rangle]$ and $A_2 = \\C[\\langle b \\rangle]$ are free.\nAn element of $A_1$ with $\\varphi = 0$ is a combination of powers $a^k$ with $k \\ne 0$, and likewise for $A_2$.\nA product of such elements with alternating indices is a combination of alternating words $a^{k_1} b^{l_1} a^{k_2} \\cdots$ with nonzero exponents.\nThese are reduced words, hence not the identity, so $\\varphi$ of the product is $0$.\nThe same argument works for any [[free product|TM7QS7P7X9]] of groups.\n\nNon-example: classically independent variables are not free.\nLet $A = L^\\infty(\\Omega, P)$ for a [[probability space|0NFPJVJJHX]] $(\\Omega, P)$, $\\varphi$ the expectation, and $x, y$ independent with mean $0$ and nonzero variance.\nThen $\\varphi(xyxy) = \\varphi(x^2)\\varphi(y^2) \\ne 0$, although the indices alternate, so the defining condition fails.\n\nOften $A$ is a C*-algebra or a von Neumann algebra and $\\varphi$ is a state, frequently a faithful trace.\nFreeness determines all mixed moments of free variables from their separate moments, as independence does classically.\n",
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