Positive Type and Positive Definite Functions on Matrix Valued Group Algebras

Abstract

Let G be a locally compact group equipped with the left Haar measure $$m_G$$ , $$M_n$$ be an $$n\times n$$ matrix with entries in $${{\mathbb {C}}}$$ and let $$L^1(G,M_n)$$ be the Banach algebra respect to the convolution products $$*$$ and $$*_\ell $$ that consists all $$M_n$$ -valued functions on G. We define the left and right positive type functions on $$(L^1(G,M_n),*)$$ and $$(L^1(G,M_n),*_\ell )$$ . Moreover, analogues to complex valued case, we construct two Hilbert spaces by the right and left positive type functions on $$(L^1(G,M_n),*)$$ and $$(L^1(G,M_n),*_\ell )$$ and we characterize the right and left positive type functions on $$(L^1(G,M_n),*)$$ and $$(L^1(G,M_n),*_\ell )$$ . We also define positive definite functions on $$(L^1(G,M_n),*)$$ and $$(L^1(G,M_n),*_\ell )$$ and we show that any continuous right (left) positive function is positive definite and vice versa.