# References for “folklore” on strong amenability of (group) C*-algebras?

[Apologies in advance for the prolixity - but I was unsure how much of the story is familiar.] $\newcommand{\ptp}{\widehat{\otimes}} \newcommand{\co}{\operatorname{co}} \newcommand{\Cst}{\operatorname{C}^*}$

Definitions. A unital $\Cst$-algebra $A$ is said to be strongly amenable if there is a bounded net $(m_i) \subset \co \{ u \otimes u^* : u\in A \hbox{ unitary}\}\subset A\ptp A$ which satisfies $\Vert a\cdot m_i - m_i \cdot a \Vert\to 0$ for each $a\in A$. Here $\co$ denotes the convex hull. A $\Cst$-algebra without unit is called strongly amenable when its unitization is. The Cuntz algebras are examples which are amenable but not strongly amenable (Rosenberg, Comm Math Phys 1977; this does not use nuclear $\implies$ amenable).

Background. These definitions are equivalent to ones made by B. E. Johnson in his 1972 Mem. AMS monograph - for good reasons, which are a bit too lengthy to go into - where some basic hereditary properties are proved and examples are given. In particular, if $G$ is a discrete amenable group then its reduced group $\Cst$-algebra is strongly amenable. (If $G$ is a discrete group whose reduced group $\Cst$-algebra is amenable, then Bunce, Proc AMS 1976, proved $G$ must be amenable; this does not use amenable $\implies$ nuclear.)

Conspicuously, Johnson does not claim that the reduced group $\Cst$-algebras of every amenable locally compact group is strongly amenable. It seems that by using material "added in proof" to his article/monograph, one can indeed show this is the case, but last time I looked at the literature a few years ago, I didn't find the argument/result stated explicitly anywhere. Hence:

Question 1. Does anyone know of a reference for "$G$ amenable $\implies$ $\Cst_r(G)$ strongly amenable" in full generality?

I seem to remember that a statement of the general case is given somewhere in Paterson's book Amenability, but the argument there seemed insufficient to me, there are technical points arising from the fact that the unitization is not the same as the multiplier algebra.

Following on from this: in the Rosenberg paper mentioned above, he also proves that the crossed product of a strongly amenable $\Cst$-algebra by the action of a discrete amenable group is again a strongly amenable $\Cst$-algebra. He mentions that the restriction to the discrete case is just to reduce the technical details, and seems to imply that the same result holds without the word "discrete".

Question 2. Again, does anyone know of a reference for this more general claim?

• @Yemon 'prolixity' - you should use any and all acronyms that you can and also the densest jargon. In fact this question would have fit into 4 sentences. ;) – David Roberts Dec 5 '11 at 1:05
• Oh, but +1 anyway :P – David Roberts Dec 5 '11 at 1:06