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

Nica, A.

J. Oper. Theory 27, No. 1, 17-52 (1992).

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

Keywords:

partially ordered group; totally ordered Abelian group; finitely supported positive forms; amenability; discrete group; \(C^*\)-algebra of the Wiener-Hopf operators; canonical conditional expectation; finitely supported positive definite functions; commutator ideal