## State space methods in operator and function theory

#### Event Title

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

#### Location

Cupples I Room 113

#### Start Date

7-19-2016 5:00 PM

#### End Date

19-7-2016 5:20 PM

#### Description

We introduce, for any set $S$, the concept of $\mf{K}$-family between two Hilbert $C^*$-modules over two $C^*$-algebras, for a given completely positive definite (CPD-) kernel $\mf{K}$ over $S$ between those $C^*$-algebras and obtain a factorization theorem for such $\mf{K}$-families. If $\mf{K}$ is a CPD-kernel and $E$ is a full Hilbert $C^*$-module, then any $\mf{K}$-family which is covariant with respect to a dynamical system $(G,\eta,E)$ on $E$, extends to a $\tilde{\mf{K}}$-family on the crossed product $E \times_\eta G$, where $\tilde{\mf{K}}$ is a CPD-kernel. Several characterisations of $\mf{K}$-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 $\mf{K}$-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 $\mf{K}$-families.

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COinS

Jul 19th, 5:00 PM Jul 19th, 5:20 PM

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

Cupples I Room 113

We introduce, for any set $S$, the concept of $\mf{K}$-family between two Hilbert $C^*$-modules over two $C^*$-algebras, for a given completely positive definite (CPD-) kernel $\mf{K}$ over $S$ between those $C^*$-algebras and obtain a factorization theorem for such $\mf{K}$-families. If $\mf{K}$ is a CPD-kernel and $E$ is a full Hilbert $C^*$-module, then any $\mf{K}$-family which is covariant with respect to a dynamical system $(G,\eta,E)$ on $E$, extends to a $\tilde{\mf{K}}$-family on the crossed product $E \times_\eta G$, where $\tilde{\mf{K}}$ is a CPD-kernel. Several characterisations of $\mf{K}$-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 $\mf{K}$-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 $\mf{K}$-families.