Determinants and multiplicative functionals on quaternion matrices (Q1399929)

A functional \(f:M\to N\) is called multiplicative if \(f(m_1m_2)=f(m_1)f(m_2)\) for any \(m_1,m_2\in M\). Determinants of real, complex matrices are multiplicative functionals. Let \(H\) be the quaternion division ring. The main theorem of this paper claims that if the functional \(f:M_n(H)\to H\) (\(n\geq 2\)) is multiplicative, then there exists a unique multiplicative functional \(g:\mathbb{R}^+\cup\{0\}\to H\), such that \(f=g\circ f_0\) where \(f_0: M_n(H)\to\mathbb{R}^+\cup\{0\}\) is defined by \(f_0(A)=\sqrt{\det\chi_A}\). As a consequence, there does not exist any extension of the determinants of real matrices to quaternion matrices as a multiplicative functional. Moreover, in some sense, the definition of determinants for quaternion matrices is unique.