Centralizing traces with automorphisms on triangular algebras
Let \(R\) be a \(2\)-torsion-free commutative ring with identity, and \(T=\binom{A\;\;M}{0\;\;B}\) a triangular algebra over \(R\), where \(A\) and \(B\) be unital algebras over \(R\) and \(M\) an \((A,B)\)-bimodule. Set \(P=\binom{1\;\;0}{0\;\;0}\), \(Q=I-P=\binom{0\;\; 0}{0\;\; 1'}\). Let \(\xi\) be an automorphism of \(T\) such that \(\xi(PTP)=PTP\), \(\xi(PTQ)=PTQ\) and \(\xi(QTQ)=QTQ\). In this case, \(\xi\) can be written as follows: \[ \xi:\binom{a \;\; m}{0 \; \; b}\mapsto \binom{f_{\xi}(a)\;\; \nu_{\xi}(m)}{0 \;\; g_{\xi}(b)}, \] where \(f_{\xi}\), \(g_{\xi}\) are automorphisms of \(A\), \(B\), respectively, and \(\nu_{\xi}:M\to M\) is an \(R\)-linear bijective which satisfies \(\nu_{\xi}(am)=f_{\xi}(a)\nu_{\xi}(m)\) and \(\nu_{\xi}(mb)=\nu_{\xi}(m)g_{\xi}(b)\), for all \(a\in A\), \(b\in B\), \(m\in M\). In this article, the authors first show that each \(\xi\)-commuting trace \(\Im_{q}:T\to T\) of an \(R\)-bilinear mapping \(q:T\times T\to T\) is proper if it satisfies 1.each \(f_{\xi}\) (resp., \(g_{\xi}\)-commuting) linear mapping on \(A\) (resp., \(B\)) is proper, 2. \(Z_{f_{\xi}}(A)\not=A\) and \(Z_{g_{\xi}}(B)\not=B\), and 3. for any \(a\in A\) and \(b\in B\), \(aMb=0\) implies that \(a=0\) or \(b=0\). Now, \(\Im_{q}\) being proper means that there are an \(R\)-linear mapping \(\mu:T\to Z_{\xi}(T)\), a trace \(\nu:T\to Z_{\xi}(T)\) of certain bilinear mapping, and a \(z\in Z_{\xi}(T)\) such that \(\Im_{q}(x)=za^{2}+\mu(a)a+\nu(a)\) for all \(a\in T\). Using this result, with the further assumption that \(T\) is partible, so \(\xi=\phi_{r}\overline{\xi}\), where \(r\in T\), \(\phi_{r}\) the inner automorphism induced by \(r\), and \(\overline{\xi}\) an automorphism of \(T\) satisfying \(\overline{\xi}(PTP)=PTP\), \(\overline{\xi}(PTQ)=PTQ\), \(\overline{\xi}(QTQ)=QTQ\), such that the above conditions hold, it is shown that each \(\xi\)-commuting trace \(\Im_{q}:T\to T\) of \(q\) is proper. The \(\xi\)-centralizing traces of bilinear mappings on \(T\) are then characterized.
scientific article; zbMATH DE number 6865403
scientific article