# Need a reference of a fact given in B. Blackadar's Operator Algebras

I am reading Blackadar's book on Operator algebras. In $$\Pi 9.6.5$$ Blackader says that

Maximal Tensor products commute with arbitrary limits.

In the same book the proof of this fact is not given.In the literature also i could not find a proof of this. Can someone give a sketch or reference for the proof of the same result?

Thanks in advance

• Can you give a reference for what Blackadar means by "limits" here? Is this "limits" in the sense of category theory? Naively I would expect them to work better with "colimits" – Yemon Choi Feb 8 at 13:07
• @YemonChoi: By limits he meant the directed limits. Probably the same holds for colimits also but I am not sure. – Math Lover Feb 9 at 7:01

## 1 Answer

I think it goes like this (I am only dealing with the unital case, I am not exactly sure how this works in the non-unital case).

Let $$(A_{i})_{i\in I}$$ be an inductive system of unital $$C^{\ast}$$-algebras with unital connecting homomorphisms $$f_{ij}: A_{i} \to A_{j}$$ and let's denote the limit by $$A$$. If $$B$$ is another unital $$C^{\ast}$$-algebra, we would like to show that the inductive limit of $$(A_{i}\otimes_{\rm{max}} B)_{i\in I}$$ is isomorphic to $$A\otimes_{\rm{max}} B$$. In order to do so, we will show that $$A\otimes_{\rm{max}}B$$ has the required universal property, i.e. any compatible family of $$\ast$$-homomorphisms from $$A_{i}\otimes_{\rm{max}} B$$ to a unital $$C^{\ast}$$-algebra $$C$$ gives rise to a $$\ast$$-homomorphism from $$A\otimes_{\rm{max}} B$$. Note now that, by the universal property of the maximal tensor product, a $$\ast$$-homomorphism $$\varphi_{i}: A_{i}\otimes_{\rm{max}} B \to C$$ is given by $$\varphi_{i} = \theta_{i}\cdot \psi_{i}$$, where $$\theta_{i}:A_{i} \to C$$ and $$\psi_{i}: B \to C$$ are $$\ast$$-homomorphisms with commuting ranges. As connecting maps are of the form $$f_{ij}\otimes \rm{Id}$$, we can check that $$\psi_{i}=\psi_{j}$$, which we call $$\psi$$ from now on, and $$\theta_{i} = \theta_{j}\circ f_{ij}$$. Since $$A$$ is the limit of $$A_i$$'s, we get a $$\ast$$-homomorphism $$\theta:A\to C$$. As the union of ranges of $$A_i$$'s inside $$A$$ is dense, this $$\ast$$-homomorphism has range commuting with the range of $$\psi$$, so we get a $$\ast$$-homomorphism $$\varphi: A\otimes_{\rm{max}} B \to C$$ given by $$\varphi(x\otimes y):= \theta(x)\psi(y)$$.

• Sorry can you give a reference for universal property of maximal tensor product? Thank you very much! – Math Lover Feb 9 at 7:03
• It really follows from the definition: the norm in the maximal tensor product is computed as the supremum of the norms under pairs of $\ast$-homomorphisms with commuting ranges. On the other hand, if we are given a $\ast$-homomorphism from the tensor product $A\otimes B$ (of unital algebras), we get a pair of homomorphisms with commuting ranges by restricting to $A\otimes \mathrm{1}$ and $1\otimes \mathrm{B}$. – Mateusz Wasilewski Feb 11 at 8:43
• Thank you. For non unital case do we need to proceed by adjoining identity or some other way? – Math Lover Feb 11 at 10:09
• I haven't checked the details, but I suppose so. One point which might be important, is that a non-unital algebra is an ideal in its unitisation and inclusions of ideals interact nicely with the maximal tensor product (which is not the case for arbitrary inclusions). – Mateusz Wasilewski Feb 11 at 10:35