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

Question

In the survey article by Maes and Van Daele in the beginning of section 5, after Def. 5.1, the following claim is made (I will use different notation here, but wanted to give the reference nonetheless):

Let $H$ be a Hilbert Space, $X$ be a compact hausdorff space and $K(H)$ denote the compact operators on H. Given any $C^\ast$-algebra $A$, we denote its multiplier algebra by $M(A)$. Then we can identify elements in $M(K(H)\otimes C(X)\otimes C(X))$ with strictly continuous functions from $X\times X$ to $B(H)$.

I have two questions regarding this:

1. The way that I understand this is that we identify $C(X)\otimes C(X)$ with $C(X\times X)$ (which I'm fine with) and then, somehow, get that $M(K(H) \otimes C(X\times X)\simeq M(K(H))\otimes C(X\times X)$. Why does that hold? Is it true in general that for any $C^\ast$-algebra $A$, we have $M(A\otimes C(X))\simeq M(A)\otimes C(X)$? Or is this something more specific about the compact operators? Am I correct in assuming that this is the intended route and we conclude the claim by identifying $M(K(H))$ with $B(H)$ (and $M(K(H))\otimes C(X)$ with the continuous functions from $X$ to $M(K(H))$, but that is something that I at least know how to do)?
2. I have read in several places that the multiplier algebra of $K(H)$ is $B(H)$ equipped with the $\sigma$-stop$^\ast$-topology. What is a good reference for that? From my understanding the multiplier algebra of a $C^\ast$-algebra $A$ should again be a $C^\ast$-algebra with respect to the norm $$\lVert \cdot\rVert_{M(A)} \colon = \sup_{a\in A} \max\{l_a(\cdot), r_a(\cdot)\},$$ where $l_a(x)\colon= \lVert xa\rVert\lVert a\rVert$ and $r_a(x)\colon=\lVert ax\rVert\lVert a\rVert$. So what exactly do people mean when they say that we can identify $B(H)$ equipped with the with the multiplier algebra of $K(H)$? They certainly cannot be isomorphic as $C^\ast$-algebras, as that would imply equality of the two norms by uniqueness of norms that turn an algebra into a $C^\ast$-algebra. What properties does this identification of $M(K(H))$ and $B(H)$ preserve? In what sense can they be identified?

I'd be extraordinarily happy for any good reference on the topic that allows me to digest and understand what is going on in the Maes and Van Daele article. Thanks in advance!

Answer 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!