Crossed Products by Abelian Semigroups via Transfer Operators

We propose a generalisation of Exel’s crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler’s theory of Toeplitz and Cuntz-Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz-Pimsner algebras of product systems. The crossed product of a C-algebra A by an endomorphism α was first constructed by Cuntz [7] as a corner in an ordinary group crossed product. Since then, other authors have proposed definitions of such a crossed product mainly in terms of universal properties of representations of A and the unital, additive semigroup N, cf. [19, 22, 2, 18]. The construction introduced by Exel [8] adds to the pair (A, α) a new ingredient, namely a transfer operator, which is a positive continuous linear map L : A→ A satisfying L(α(a)b) = aL(b) for all a, b ∈ A. The resulting crossed product A⋊α,L N generalises the previous constructions from [19, 22, 2, 18], see also the more recent [9]. In Stacey’s theory of crossed products by single endomorphisms [22], the extension to other semigroups brought in a large number of new and interesting examples and applications, see for example [1, 2, 13, 14]. Here we seek an extension of Exel’s theory, and we are motivated by the following situation. For a compact abelian group Γ there is always an action β : N → End(C(Γ)) of the multiplicative semigroup N given by βn(f)(s) = f(s). What we notice is that under certain finiteness conditions there is an action of transfer operators Ln for βn, see Proposition 4.1, and then we wish to form and analyse a crossed product C(Γ)⋊β,L N. It is our goal here to adapt Exel’s definition to the case of actions of abelian semigroups of endomorphisms and corresponding transfer operators. As applications, we take a closer look at the motivating example. Our construction allows us to view in a unified way the crossed product by two rather different types of actions, see Theorem 4.7, and for the motivating example we use this to show that in Date: February 15, 2005. 1991 Mathematics Subject Classification. 46L55. Research partly supported by The Carlsberg Foundation and the Centre for Advanced Study, Oslo.