## Rocky Mountain Journal of Mathematics

### $\mathfrak K$-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.

#### Article information

Source
Rocky Mountain J. Math., Volume 47, Number 3 (2017), 789-816.

Dates
First available in Project Euclid: 24 June 2017

https://projecteuclid.org/euclid.rmjm/1498269811

Digital Object Identifier
doi:10.1216/RMJ-2017-47-3-789

Mathematical Reviews number (MathSciNet)
MR3682149

Zentralblatt MATH identifier
1381.46053

#### Citation

Dey, Santanu; Trivedi, Harsh. $\mathfrak K$-families and CPD-H-extendable families. Rocky Mountain J. Math. 47 (2017), no. 3, 789--816. doi:10.1216/RMJ-2017-47-3-789. https://projecteuclid.org/euclid.rmjm/1498269811

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