\(R\)-diagonal elements and freeness with amalgamation (Q2715674)

Let \(( {\mathcal A}, \varphi)\) be a \(*\)-probability space, where \({\mathcal A}\) is a unital \(*\)-algebra over \({\mathbb C}\), and \(\varphi\) : \({\mathcal A} \rightarrow {\mathbb C}\) is a linear functional normalized by \(\varphi (I) = 1\) with the unit \(I\) of \({\mathcal A}\), such that \(\varphi(a^*)\) \(=\) \({\overline \varphi}(a)\) for every \(a \in {\mathcal A}\). An element \(a \in {\mathcal A}\) is considered as a non-commutative random variable and \(\varphi(a)\) is called the expectation of \(a\). \(({\mathcal A}, \varphi)\) is said to be tracial if \(\varphi\) has the trace property : \(\varphi(ab)\) \(=\) \(\varphi(ba)\) for every \(a, b\) \(\in\) \({\mathcal A}\). The expectations of words made with \(a\) and \(a^*\) is called \(*\)-moments of \(a\). If \(({\mathcal A}, \varphi)\) and \(({\mathcal B}, \psi)\) are \(*\)-probability spaces, two elements \(a \in {\mathcal A}\), \(b \in {\mathcal B}\) are identically \(*\)-distributed if their \(*\)-moments coincide : \(\varphi( a^{s_1}\) \(\cdots\) \(a^{s_n})\) \(=\) \(\psi( b^{s_1}\) \(\cdots\) \(b^{s_n})\) for every \(n \geq 1\) and \(s_1, \dots, s_n\) \(\in\) \(\{ 1, * \}\). Roughly speaking, the name \(R\)-diagonal refers to elements which have a factorization of the form : \(a = u p\) \(( u, p\) \(\in\) \({\mathcal A})\) where \(u\) is a unitary such that \(\varphi( u^n)\) \(=0\) for every \(n \geq 1\), and the sets \(\{ u, u^* \}\) and \(\{ p, p^* \}\) are free in the sense of \textit{D. V. Voiculescu, K. J. Dykema} and \textit{A. Nica} [Free Random Variables, CRM monograph Series 1, Am. Math. Soc., Providence (1992; Zbl 0795.46049)]. The concept of \(R\)-diagonal element was introduced in \textit{A. Nica} and \textit{R. Speicher} [Fields Inst. Commun. 12, 149-188 (1997; Zbl 0889.46053)] in connection with applications to several problems in free probability. NEWLINENEWLINENEWLINEIn this paper the authors describe a new approach to \(R\)-diagonality, relying on freeness with amalgamation. The class of \(R\)-diagonal elements is enlarged to contain examples living in non-tracial \(*\)-probability spaces, such as the generalized circular elements of \textit{D. Shlyakhtenko} [Pac. J. Math. 177, No. 2, 329-368 (1997; Zbl 0882.46026)]. Let \(M_2({\mathcal A})\) be the algebra of \(2 \times 2\) matrices over \({\mathcal A}\) and \(D\) is the unital subalgebra of \(M_2( {\mathbb C} I)\) \(\subset\) \(M_2({\mathcal A})\) with NEWLINE\[NEWLINE D : = \left\{ \left(\begin{matrix}\alpha I & 0 \\ 0 & \lambda I \end{matrix}\right) \Biggl|\alpha, \lambda \in {\mathbb C} \right\}, \quad M_2( {\mathbb C} I) := \left\{\left(\begin{matrix}\alpha I & \beta I \\ \gamma I & \lambda I \end{matrix}\right) \Biggl|\alpha, \beta, \gamma, \lambda \in {\mathbb C} \right\}. NEWLINE\]NEWLINE The main theorem asserts: the following five conditions on \(a \in {\mathcal A}\) are equivalent. The element \(a\) is said to be \(R\)-diagonal if it satisfies one (hence all) of these conditions.NEWLINENEWLINENEWLINE\(1^{\circ}\) (Condition on \(*\)-moments) \( \varphi( P_{ i_1 i_2 ; k_1} (a) P_{i_2 i_3; k_2}(a) \cdots P_{i_n i_{n+1} ; k_n }(a)) = 0 \) for every \(n \geq 1\), \(i_1, \dots, i_{n+1} \in \{ 1,2 \}\) and \(k_1, \dots, k_n\) \(\geq 1\), where \(P_{ij ; k}(a)\) \(\in\) \({\mathcal A}\) with NEWLINE\[NEWLINE P_{1 1 ; k }(a) = a^* (a a^*)^{k-1}, \quad P_{1 2; k}(a) = ( a^* a)^k - \varphi( (a^* a)^k) I, NEWLINE\]NEWLINE NEWLINE\[NEWLINE P_{2 1 ; k}(a) = ( a a^*)^k - \varphi( (a a^*)^k) I, \quad \text{and} \quad P_{2 2 ; k}(a) = a ( a^* a)^{k-1}. NEWLINE\]NEWLINE \(2^{\circ}\) (Sufficient invariance condition) There exist an enlargement \(( \widetilde{{\mathcal A}}, \widetilde{\varphi})\) of \(( {\mathcal A}, \varphi)\) and a unitary \(u \in \widetilde{{\mathcal A}}\) such that (i) \(\{ u, u^* \}\) is free from \(\{ a, a^* \}\) ; (ii) \(\widetilde{\varphi}(u) =0\) ; (iii) \(a\) and \(ua\) are identically \(*\)-distributed. NEWLINENEWLINENEWLINE\(3^{\circ}\) (Necessary invariance condition) For every enlargement \(( \widetilde{{\mathcal A}}, \widetilde{\varphi})\) of \(( {\mathcal A}, \varphi)\) and every unitary \(u \in \widetilde{{\mathcal A}}\) such that \(\{ u, u^* \}\) is free from \(\{ a, a^* \}\), one has that \(a\) and \(ua\) are identically \(*\)-distributed. NEWLINENEWLINENEWLINE\(4^{\circ}\) (Condition on non-crossing cumulants) For the family NEWLINE\[NEWLINE \{ \kappa_{ ( s_1, \dots, s_n)} ( a, a^*) \mid n \geq 1, \;s_1, \dots, s_n \in \{ 1, * \}\} NEWLINE\]NEWLINE of non-crossing cumulants of \(a\) and \(a^*\), we have \(\kappa_{( s_1, \dots, s_n)}( a, a^*)\) \(=0\) whenever \(( s_1\), \(\dots\), \(s_n)\) is not of the form \((1, *, 1, *\), \(\dots\), \(1, *)\) or \((*, 1, *, 1\), \(\dots\), \(*, 1)\). NEWLINENEWLINENEWLINE\(5^{\circ}\) (Condition using freeness with amalgamation) Consider the conditional expectation \(E\) : \(M_2({\mathcal A})\) \(\rightarrow\) \(D\) given by the formula : NEWLINE\[NEWLINE E \left( \left( \begin{matrix} a & b \\ c& d \end{matrix} \right) \right) = \left( \begin{matrix}\varphi(a) I & 0 \\ 0 & \varphi(d) I\end{matrix} \right) NEWLINE\]NEWLINE for \(a, b, c, d\) \(\in {\mathcal A}\). Then the matrix NEWLINE\[NEWLINE A : = \left( \begin{matrix} 0 & a \\ a^* & 0 \end{matrix} \right) \in M_2({\mathcal A}) NEWLINE\]NEWLINE is free from \(M_2( {\mathbb C} I)\) with amalgamation over \(D\).