A locally compact group $G$ has Kazhdan's property (T) if the trivial representation $1_G:G\to\mathbb{C}$, $1_G(x) = 1$ for all $x\in G$, is isolated in $\hat{G}$ with the Fell topology. Bekka took this concept from the LCG setting to the $C^*$-algebras. The snippet below with the definition of property (T) for $C^*$-algebras is taken from Brown2006.

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Brown refers to Bekka2006 for the definition of property (T) although the definitions in Brown2006 and in Bekka2006 are not verbatim the same.

Question 0: Is there a written proof in the literature (a monograph, book, or an expository text on the subject) showing that these two definitions are indeed equivalent?

The snippet below is from Bekka2006.

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This is the first and the only instance that considers property (T) for general Banach algebras to the best of my knowledge.

Question 1: Could you share the links of the literature on property (T) (not only for $C^*$-algebras but also) for general Banach algebras? If you know that there is none, writing this down to the comments below is also a useful input for the amateur who posed this question.

It is surprising to me that there is a vast literature for amenability for Banach algebras (which had been taken from LCG setting as well), but relatively scarce resources for property (T).



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