# Multiplicative Domains

Posted on 4th August 2019

Reading Ng, Viselter, Amenability of locally compact quantum groups and their unitary co-representations Arxiv I was reminded by a result/technique that I often forget: that of the multiplicative domain of a completely positive map. The following result is due to M.D. Choi (though I like the presentation in Paulsen's book).

Let $$\phi:A\rightarrow B$$ be a unital completely positive (UCP) map between unital $$C^\ast$$-algebras. If $$a\in A$$ with $$\phi(a)^\ast\phi(a) = \phi(a^\ast a)$$ then already $$\phi(ba) = \phi(b)\phi(a)$$ for any $$b$$. If this is the "right hand version" then there is a left hand version, and a bimodule version.

A nice application is the following: suppose that $$u\in A$$ is unitary, and that we know that $$\phi(u)$$ is also unitary. Then $$\phi(au) = \phi(a)\phi(u)$$ for any $$a\in A$$. This follows, of course, because $$u^\ast u=1$$ and $$\phi(u)^\ast\phi(u)=1$$ so we have equality as $$\phi$$ is unital.

The application of Chi-Keung and Ami is to multiplicative unitaries which allows them to give an incredibly short proof that amenability of representations respected weak containment.

I want to just say a couple of words about why this is perhaps not too surprisingly. (More than a couple of words is maybe not warranted, as the proof itself is not so long, using the Schwarz inequality). Any UCP map is of the form $$\phi(a) = V^\ast \pi(a)V$$ where $$\pi:A\rightarrow B(H)$$ is a $$\ast$$-representation, and $$V:K\rightarrow H$$ is an isometry (here I assume $$B\subseteq B(K)$$). If $$u\in A$$ is unitary then so is $$\pi(u)$$, and so that $$V^\ast \pi(u) V$$ is unitary is a strong condition:

• $$\pi(u)$$ must restrict to a unitary on the image of $$V$$;
• thus $$VV^\ast\pi(u)V = \pi(u)V$$;
• then $$\phi(a)\phi(u) = V^\ast\pi(a) VV^\ast\pi(u) V=V^\ast \pi(au)V = \phi(au)$$.

Coda: Looking at the Stinespring dilation is often profitable.