## $$C^*$$-algebras generated by isometries and Wiener-Hopf operators.(English)Zbl 0809.46058

For a pair consisting of a discrete group $$G$$ and its subsemigroup $$P$$ satisfying $$P\cap P^{-1}= \{e\}$$ (called a partially ordered group) such as $$G= \mathbb{Z}$$, $$P= \mathbb{N}$$, the $$C^*$$-algebra $$W(G,P)$$ generated by the reduction of the left regular representation of $$G$$ to $$\ell^ 2(P)$$ is called the $$C^*$$-algebra of the Wiener-Hopf operators. The author studies the amenability of $$(G,P)$$ when any $$x_ 1,\dots,x_ n$$ in $$G$$ (for any $$n$$), which have a common upper bound in $$P$$, have the least common upper bound in $$P$$. In this case $$(G,P)$$ is said to be lattice- ordered and $$W(G,P)$$ is generated by isometries (§2.4).
The author defines the universal $$C^*$$-algebra $$C^*(G,P)$$, (§4.1) which has a canonical $$*$$-homomorphism $$\pi_ W$$ onto $$W(G,P)$$, and $$(G,P)$$ is defined to be amenable if and only if $$\pi_ W$$ is one-to- one. (§4.2). This is shown to be equivalent to each of the following conditions:
(1) A canonical conditional expectation from $$C^*(G,P)$$ to its specific Abelian $$C^*$$-subalgebra (a diagonal subalgebra) is faithful.
(2) Finitely supported positive forms are weak $$*$$ dense in all positive forms on $$C^*(G,P)$$. The amenability is implied by the existence of a net of finitely supported positive definite functions on $$PP^{-1}$$ converging pointwise to 1.
$$(G,P)$$ is amenable if $$G$$ is amenable or if $$P$$ is Abelian. $$(F_ n, SF_ n)$$ is amenable for the free group $$F_ n$$ of $$n$$ generators and the semigroup $$SF_ n$$ generated by free generators.
As an application, the author proves that $$W(G,P)$$ contains the compact operators if and only if there exists a finite subset of $$P\backslash \{e\}$$ which contains a lower bound for any element of $$P\backslash \{e\}$$, a condition which holds if $$P$$ is finitely generated. For another application, the following conditions for a totally ordered Abelian group $$(G,P)$$ are shown to be equivalent.
(1) $$P$$ is Archimedean.
(2) Any two non-unitary representations by isometries of $$P$$ generate canonically isomorphic $$C^*$$-algebras.
(3) The commutator ideal of $$W(G,P)$$ is simple.
Reviewer: H.Araki (Kyoto)

### MSC:

 46L05 General theory of $$C^*$$-algebras 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators