Toeplitz algebras, subnormal tuples and rigidity on reproducing C[z1,…,zd]-modules

This paper mainly considers Toeplitz algebras, subnormal tuples and rigidity concerning reproducing C[z1,…,zd]-modules. By making use of Arvesonʹs boundary representation theory, we find there is more rigidity in several variables than there is in single variable. We specialize our attention to reproducing C[z1,…,zd]-modules with U-invariant kernels by examining the spectrum and the essential spectrum of the d-tuple {Mz1,…,Mzd}, and deducing an exact sequence of C∗-algebras associated with Toeplitz algebra. Finally, we deal with Toeplitz algebras defined on Arveson submodules and rigidity of Arveson submodules.