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Amenability of Hopf von Neumann algebras and Kac algebras. (English) Zbl 0896.46041
A 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.

##### MSC:
 46L10 General theory of von Neumann algebras 47L50 Dual spaces of operator algebras
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