Amenability of Hopf von Neumann algebras and Kac algebras.

*(English)*Zbl 0896.46041A Hopf-von Neumann algebra is a pair \((M,\Gamma)\), where \(M\) is a \(W^*\)-algebra and \(\Gamma\) is a unital injective normal *-homomorphism from \(M\) into the \(W^*\)-tensor product \(M\overline\otimes M\) such that
\[
(\Gamma\otimes id)\circ\Gamma= (id\otimes\Gamma)\circ \Gamma.
\]
The Hopf-von Neumann algebra \((M,\Gamma)\) is said to be co-involutive if there exists a unital normal anti-automorphism \(\kappa\) on \(M\) such that
\[
(\kappa\otimes\kappa)\circ \Gamma= \sigma\circ\Gamma\circ \kappa,
\]
where, for \(a\) and \(b\) in \(M\),
\[
\sigma(a\otimes b)= b\otimes a.
\]
A Kac algebra is a quadruple \({\mathbf K}\) equal to \((M,\Gamma, \kappa,\phi)\), where \((M,\Gamma,\kappa)\) is a co-involutive Hopf-von Neumann algebra and \(\phi\) is a left Haar weight on \(M\).

Any \(W^*\)-algebra \(M\) and its predual \(M_*\) form operator spaces and the operator projective tensor product \(M_*\widehat\otimes M_*\) is completely isometric to the predual \((M\overline\otimes M)_*\) of \(M\overline\otimes M\). It follows from the relation above that the mapping \(\Gamma_*\) from \(M_*\widehat\otimes M_*\) to \(M_*\) is an associative completely contractive multiplication on \(M_*\) and hence that \(M_*\) is a Banach algebra the multiplication in which is completely contractive. If \((M,\Gamma)\) is co-involutive then the mapping \(\omega\to \omega^0\) defined, for \(a\) in \(M\), by \[ \omega^0(a)= \omega^*(\kappa(a))= \overline{\omega(\kappa(a^*))} \] is an involution on \(M_*\). In particular, if \({\mathbf K}\) is a Kac algebra then \(M_*\) is an involutive completely contractive Banach algebra. Moreover, \({\mathbf K}\) is said to be discrete if \(M_*\) possesses a multiplicative unit. Every Kac algebra \({\mathbf K}\) possesses a canonically defined dual Kac algebra, denoted by \(\widehat{\mathbf K}\).

The Kac algebra \({\mathbf K}\) is said to be operator amenable if, for every operator \(M_*\)-bimodule \(V\), every completely bounded derivation from \(A\) to \(V^*\) is inner, is said to be Voiculescu amenable if there exists a state \(\nu\) of \(M\) such that, for all \(\omega\) in \(M_*\), \[ (\omega\otimes \nu)\circ\Gamma= \omega(1)\nu, \] and is said to be strongly Voiculescu amenable if there exists a net \((\nu_j)\) of normal states in \(\widehat M_*\) which is a left approximate identity for \(\widehat M_*\).

The author shows that the following conditions on a discrete Kac algebra \({\mathbf K}\) are equivalent: \({\mathbf K}\) is operator amenable; \({\mathbf K}\) is Voiculescu amenable; \(\widehat M\) is hyperfinite; \({\mathbf K}\) is strongly Voiculescu amenable; \(\widehat{\mathbf K}\) is operator amenable; \(\widehat M_*\) has a bounded approximate identity.

This result fills in several large gaps in the theory of discrete Kac algebras.

Any \(W^*\)-algebra \(M\) and its predual \(M_*\) form operator spaces and the operator projective tensor product \(M_*\widehat\otimes M_*\) is completely isometric to the predual \((M\overline\otimes M)_*\) of \(M\overline\otimes M\). It follows from the relation above that the mapping \(\Gamma_*\) from \(M_*\widehat\otimes M_*\) to \(M_*\) is an associative completely contractive multiplication on \(M_*\) and hence that \(M_*\) is a Banach algebra the multiplication in which is completely contractive. If \((M,\Gamma)\) is co-involutive then the mapping \(\omega\to \omega^0\) defined, for \(a\) in \(M\), by \[ \omega^0(a)= \omega^*(\kappa(a))= \overline{\omega(\kappa(a^*))} \] is an involution on \(M_*\). In particular, if \({\mathbf K}\) is a Kac algebra then \(M_*\) is an involutive completely contractive Banach algebra. Moreover, \({\mathbf K}\) is said to be discrete if \(M_*\) possesses a multiplicative unit. Every Kac algebra \({\mathbf K}\) possesses a canonically defined dual Kac algebra, denoted by \(\widehat{\mathbf K}\).

The Kac algebra \({\mathbf K}\) is said to be operator amenable if, for every operator \(M_*\)-bimodule \(V\), every completely bounded derivation from \(A\) to \(V^*\) is inner, is said to be Voiculescu amenable if there exists a state \(\nu\) of \(M\) such that, for all \(\omega\) in \(M_*\), \[ (\omega\otimes \nu)\circ\Gamma= \omega(1)\nu, \] and is said to be strongly Voiculescu amenable if there exists a net \((\nu_j)\) of normal states in \(\widehat M_*\) which is a left approximate identity for \(\widehat M_*\).

The author shows that the following conditions on a discrete Kac algebra \({\mathbf K}\) are equivalent: \({\mathbf K}\) is operator amenable; \({\mathbf K}\) is Voiculescu amenable; \(\widehat M\) is hyperfinite; \({\mathbf K}\) is strongly Voiculescu amenable; \(\widehat{\mathbf K}\) is operator amenable; \(\widehat M_*\) has a bounded approximate identity.

This result fills in several large gaps in the theory of discrete Kac algebras.

Reviewer: C.M.Edwards (Oxford)