Left multipliers of reproducing kernel Hilbert $C^*$-modules and the Papadakis theorem

TL;DR

This paper extends the theory of reproducing kernel Hilbert $C^*$-modules by removing self-duality, exploring tensor products, left multipliers, and the Papadakis theorem in this context.

Contribution

It introduces a modified definition of $RKHC^*M$ without self-duality, studies their tensor products, multipliers, and extends the Papadakis theorem.

Findings

A new $RKHC^*M$ can be constructed via left multipliers.

The reproducing kernel of tensor products of $RKHC^*M$s is identified.

The Papadakis theorem is extended to $RKHC^*M$ setting.

Abstract

We give a modified definition of a reproducing kernel Hilbert $C^*$-module (shortly, $RKHC^*M$) without using the condition of self-duality and discuss some related aspects; in particular, an interpolation theorem is presented. We investigate the exterior tensor product of $RKHC^*M$s and find their reproducing kernel. In addition, we deal with left multipliers of $RKHC^*M$s. Under some mild conditions, it is shown that one can make a new $RKHC^*M$ via a left multiplier. Moreover, we introduce the Berezin transform of an operator in the context of $RKHC^*M$s and construct a unital subalgebra of the unital $C^*$-algebra consisting of adjointable maps on an $RKHC^*M$ and show that it is closed with respect to a certain topology. Finally, the Papadakis theorem is extended to the setting of $RKHC^*M$, and in order for the multiplication of two specific functions to be in the Papadakis $RKHC^*M$, some conditions are explored.