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.

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