Induced C*-algebras, coactions and equivariance in the symmetric imprimitivity theorem

Abstract

The symmetric imprimitivity theorem provides a Morita equivalence between two crossed products of induced C*-algebras and includes as special cases many other important Morita equivalences such as Green's imprimitivity theorem. We show that the symmetric imprimitivity theorem is compatible with various inflated actions and coactions on the crossed products.

A categorical approach to imprimitivity theorems for 𝐶*-dynamical systems

“…These are used repeatedly: a coaction on a bimodule X, for example, is by definition a homomorphism of X into the multiplier bimodule M (X ⊗ C * (G)). Our treatment is similar to that of imprimitivity bimodules in [20]. Section 1.3 is about the balanced tensor products which are used to define the composition of morphisms; we need to know in particular how this process extends to multipliers.…”

“…For imprimitivity bimodules, it is handy to INTRODUCTION recognize that if L(X) is the linking algebra of X, then the bimodule crossed product X × G embeds as the top right corner of L(X) × G, and we have the important relation L(X) × G = L(X × G) almost by definition. We should mention that defining these crossed products and establishing their properties has been done before; see [2], [7], [6], [20], and [30].…”

Regularity of induced representations and a theorem of Quigg and Spielberg

“…the only difference between (2) and 3is the location of the integral with respect to ⊗ B h. But the integrands in both formulas are continuous and compactly supported, so there is no difficulty verifying that they have the same inner product with every vector of the form y ⊗ B k ∈ Z 0 H, and hence must be equal. We deduce that…”

“…In [2, section 1] Echterhoff and Raeburn consider the special case where H and K are subgroups of the same locally compact group G and P = G. They construct a pair of regular representations of the induced systems (Ind(C, σ), H) and (Ind(C, τ ), K), and show that these induce to each other via the equivalence of the symmetric imprimitivity theorem [2, theorem 1.4]. We can use our Theorem 1 to see directly that the induced representations are regular: in the notation of [2], just define µ: s, k, h)), and then µ is a representation of C 0 (G, D) such that the restriction ofμ to the induced algebra is Ind G H (ρ 1 ⊗ 1 ⊗ 1).…”

“…We can use it to see, albeit somewhat indirectly, that regular representations themselves nearly always induce to regular representations (see Corollary 7), and it gives a new proof of the main theorem of [5, section 4] which avoids a complicated argument involving a composition of crossed-product Morita equivalences (see Corollary 11). It also sheds light on constructions in [2] and [8], which, in various special cases of the symmetric imprimitivity theorem, yield pairs of regular representations which induce to each other (see Remarks 10).…”