Is the multiplier algebra $M(K(H)\otimes C(X))$ isomorphic to $M(K(H))\otimes C(X)$?

Solution 1:

Firstly, $M(K(H)) \cong B(H)$ is definitely true. This is an identification of $C^*$-algebras. I don't understand your objection about the norms.

I think you are misunderstanding what Maes and Van Daele try to say in their paper. The topology considered on $B(H)$ is important here. While you consider $C(X,B(H))$ where $B(H)$ has the norm-topology, they look at $C(X,B(H))$ where $B(H)$ has the strong topology. Moreover, you should also take into account that they consider a unitary representation $$u: X \to B(H)$$ and for such a representation, one can prove that $u$ is strongly continuous if and only if it is strong$^*$-continuous if and only if it strictly continuous, where the strict topology on $B(H)$ comes from the isomorphism $B(H)\cong M(K(H)).$ This being said, the correct isomorphism is $$M(K(H)\otimes C(X)) \cong C^{str}_b(X, B(H)).$$

Here, $B(H)$ is considered with the strict topology coming from the isomorphism $B(H)\cong M(K(H))$ and $C_b^{str}(X,B(H))$ denotes the continuous bounded functions from $X$ to $B(H)$ (with the strict topology). Proving this isomorphism takes some work, and probably deserves a post on this site on its own.

The map $$\pi: M(K(H)\otimes C(X))\to C_b^{str}(X,B(H))$$ is relatively easy to describe, if you are familiar with the fact that non-degenerate $*$-homomorphisms extend to the multiplier algebras. Consider a multiplier $M \in M(K(H)\otimes C(X)).$ Given $x \in X$, consider $$\iota \otimes \operatorname{ev}_x: K(H)\otimes C(X)\to K(H)$$ which is a surjective $*$-homomorphism, so in particular it is non-degenerate, so it extends uniquely to a unital $*$-homomorphism $$M(K(H)\otimes C(X)) \to M(K(H)) \cong B(H)$$

and thus we obtain the extended $*$-morphism $$\iota \otimes \operatorname{ev}_x: M(K(H)\otimes C(X)) \to B(H).$$ Hence, we obtain a map $$\pi_M: X \to B(H): x \mapsto (\iota \otimes \operatorname{ev}_x)(M)$$ and in this way we obtain the map $\pi: M(K(H)\otimes C(X)) \to C_b^{str}(X,B(H))$.

A good reference to understand all the stuff that I'm talking about is Lance's book "Hilbert $C^*$-modules", though the isomorphism I talk above is not proven in this book. In particular, the claim $M(K(H)) \cong B(H)$ follows from theorem 2.18 in Lance's book.

Also, don't get discouraged when reading Maes and Van Daele's paper. It is a difficult paper for beginners (I used to be in your shoes) and it contains some mistakes here and there. To supplement reading this paper, I recommend the following two references:

• Timmerman's book "An invitation to quantum groups and duality".
• Neshveyev's and Tuset's book "Compact quantum groups and their representation categories".

I would suggest that if you arrive at the part in Maes and Van Daele's paper where they discuss the contragredient representation, that you switch to Neshveyev's book. The section in the paper contains mistakes, depends on coordinates, and the treatment in Neshveyev-Tuset is very useful for reading follow-up papers (it is also coordinate-free). Timmerman's book is a good book to obtain a bigger picture, and also has more details in it than the paper sometimes, and often offers another valuable point of view.

Good luck, and don't hesitate to ask other (follow-up) questions!