×

Property \((T)\) for Kac algebras. (English) Zbl 0785.46058

A Kac algebra is a quadruplet \(\mathbb{K}= (M,\Gamma,K,\varphi)\) where \(M\) is a \(W^*\)-algebra, \(\Gamma: M\to M\otimes M\) is a normal injective homomorphism such that \((\Gamma\otimes \text{id})\circ \Gamma=( \text{id}\otimes \Gamma)\circ \Gamma\), \(K\) is an involutive skew automorphism of \(M\) such that \(\xi\circ\Gamma\circ K= (K\otimes K)\circ \Gamma\), where \(\xi(x\otimes y)= y\otimes x\) \((x,y\in M)\) and \(\varphi\) is a faithful normal semifinite weight such that
(i) \((\text{id}\otimes \varphi) (\Gamma(x))= \varphi(x).1\) for all \(x\in M_ +\);
(ii) \((\text{id}\otimes \varphi) ((1\otimes y^*) \Gamma(x))= K((\text{id}\otimes \varphi)(\Gamma(y^*) (1\otimes x))\) for all \(x\), \(y\) with \(\varphi(x^* x)<\infty\) and \(\varphi(y^* y)<\infty\);
(iii) \(\sigma_{*t}^ \varphi\circ K=K\circ \sigma_{-t}^ \varphi\) for all \(t\in\mathbb{R}\),
where \(\sigma^ \varphi\) is the modular automorphism of \(\varphi\).
A Kac algebra \(\mathbb{K}\) has property \((T)\) is the class of the trivial representation of \(M_ *\) is isolated in the spectrum of the involutive Banach algebra \(M_ *\).
This definition is coherent with the classical one by Kazhdan for groups: a locally compact group \(G\) has property \((T)\) if and only if its corresponding Kac algebra \(L^ \infty(G)\) has property \((T)\). The paper is devoted to the study of property \((T)\) for Kac algebras.
One of the main results is a partial improvement of theorem by A. Connes and V. Jones [Bull. London Math. Soc. 17, 57–62 (1985; Zbl 1190.46047)]: if \(G\) is a separable locally compact group and the von Neumann algebra \({\mathcal L}(G)\) generated by the left regular representation of \(G\) has property \((T)\) then \(G\) has property \((T)\).

MSC:

46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
47L50 Dual spaces of operator algebras

Citations:

Zbl 1190.46047
PDFBibTeX XMLCite