Abstract
The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups. We show that in many cases, isomorphisms of Cuntz semigroups that respect this additional structure can be lifted to Hopf algebra (bi)isomorphisms, up to a possible flip of the co-product. This shows that the Cuntz semigroup provides an interesting invariant of C*-algebraic quantum groups.
Highlights
In this paper, we address two problems, the second depending on the first
Drawing a parallel with the C*-algebraic Cuntz semigroup motivates the use of a coarser equivalence relation on Hilbert modules than the isomorphism relation that we first consider
The second problem we study is to decide when a similar statement holds at the level of Cuntz semirings and C*-algebraic quantum groups
Summary
Our result gives a condition for a mapping of Hilbert modules that is an isometry in a certain weak sense to be a unitary operator at the Hilbert module level This technique is used to show that the product we define respects the complicated and delicate equivalence relations of the Cuntz semigroup. The map given by taking a product with a fixed element respects the property of being a sub Hilbert module: in other words, if x is contained in y M x is contained in. In the finite-dimensional case, the Cuntz semigroup simplifies considerably, and our product reduces to the simpler product considered in [1] This product is defined by ∆∗ ( M1 ⊗ M2 ) where M1 and M2 are algebraic modules over A, and ∆∗ is the algebraic restriction of rings operation induced by the coproduct homomorphism. The connection technique does not seem applicable in our case
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