The Dirichlet property for tensor algebras (Q2865908)

An operator algebra \(\mathcal A\) is Dirichlet if \(\iota(\mathcal A)+\iota(\mathcal A)^*\) is dense in the \(C^*\)-envelope \(C^*_{\text{env}}(\mathcal A)\), where \(\iota:\mathcal A\to C^*_{\text{env}}(\mathcal A)\) is the canonical inclusion.NEWLINENEWLINEA \(C^*\)-correspondence \(X\) over a \(C^*\)-algebra \(A\) is a right Hilbert \(A\)-module together with a \(\ast\)-homomorphism \(\phi_X:A\to\mathcal L(X)\), where \(\mathcal L(X)\) denotes the set of bounded adjointable operators on \(X\). The Toeplitz-Cuntz-Pimsner algebra \(\mathcal T_X\) is the universal \(C^*\)-algebra for `all' representations of \(X\), and the tensor algebra \(\mathcal T^+_X\) is the norm-closed algebra generated by the copy of \(A\) and \(X\) in \(\mathcal T_X\).NEWLINENEWLINEIt is proved that \(\mathcal T^+_X\) is Dirichlet iff \(X\) is a Hilbert bimodule. It is also shown that several tensor algebras, important for applications, have the unique extension property.