$\mathfrak K$-families and CPD-H-extendable families

Abstract

We introduce, for any set $S$, the concept of a $\mathfrak {K}$-family between two Hilbert $C^*$-modules over two $C^*$-algebras, for a given completely positive definite (CPD-) kernel $\mathfrak {K}$ over $S$ between those $C^*$-algebras, and we obtain a factorization theorem for such $\mathfrak {K}$-families. If $\mathfrak {K}$ is a CPD-kernel and $E$ is a full Hilbert $C^*$-module, then any $\mathfrak {K}$-family which is covariant with respect to a dynamical system $(G,\eta ,E)$ on $E$, extends to a $\widetilde {\mathfrak {K}}$-family on the crossed product $E \times _\eta G$, where $\widetilde {\mathfrak {K}}$ is a CPD-kernel. Several characterizations of $\mathfrak {K}$-families, under the assumption that ${E}$ is full, are obtained, and covariant versions of these results are also given. One of these characterizations says that such $\mathfrak {K}$-families extend as CPD-kernels, between associated (extended) linking algebras, whose $(2,2)$-corner is a homomorphism and vice versa. We discuss a dilation theory of CPD-kernels in relation to $\mathfrak {K}$-families.