I was studying J. M. G. Fell’s paper *The Structure of Algebras of Operator Fields* when I encountered the concept of a correlation between two $ C^{\*} $-algebras.



> 
> **Definition.** Let $ A $ and $ B $ be $ C^{\*} $-algebras. Then an $ (A,B) $-correlation is a relation $ R \subseteq A \times B $ such that there exist another $ C^{\*} $-algebra $ C $ and surjective $ \* $-homomorphisms, $ f: A \to C $ and $ g: B \to C $, satisfying
>  $$
> \forall a \in A, ~ \forall b \in B: \quad
> (a,b) \in R \iff f(a) = g(b).
> $$
> 
> 
> 


In the same paper, another definition is given.



> 
> **Definition.** Let $ A $ and $ B $ be $ C^{\*} $-algebras. Then an $ (A,B) $-correlation is a closed $ C^{\*} $-subalgebra of the direct product $ C^{\*} $-algebra $ A \times B $ satisfying
>  $$
> \{ a \in A \mid \exists b \in B: ~ (a,b) \in R \} = A \quad \text{and} \quad
> \{ b \in B \mid \exists a \in A: ~ (a,b) \in R \} = B.
> $$
> 
> 
> 


It is very easy to see that the first definition implies the second one. However, I have difficulty in showing that the second implies the first. My difficulty lies chiefly in constructing the $ C^{\*} $-algebra $ C $ required by the first definition. At first, I thought that $ C $ would be $ (A \times B) / \langle R \rangle $, where $ \langle R \rangle $ denotes the closed two-sided ideal generated by $ R $, but this does not appear right.


I suspect that to construct $ C $, one must take the free $ \* $-algebra over the elements of $ A $ and $ B $, modulo the relations in $ A $ and in $ B $ and also the relations between elements of $ A $ and elements of $ B $ implemented by $ R $. However, I am not sure if such an object has a faithful $ C^{\*} $-algebraic representation.



A question about correlations between $ C^{*} $-algebras
It turns out that I have figured it out. Here is a full solution.




---


Suppose that the hypotheses on $ R $ in the second definition hold. Let
$$
I\_{A} \stackrel{\text{df}}{=} \{ a \in A \mid (a,0\_{B}) \in R \} \quad \text{and} \quad
I\_{B} \stackrel{\text{df}}{=} \{ b \in B \mid (0\_{A},b) \in R \}.
$$



> 
> **Claim 1:** $ I\_{A} $ and $ I\_{B} $ are **closed** two-sided ideals of $ A $ and $ B $ respectively.
> 
> 
> 


*Proof of Claim 1*


As $ R $ and $ A \times \{ 0\_{B} \} $ are $ C^{\*} $-subalgebras of $ A \times B $, we see that $ R \cap (A \times \{ 0\_{B} \}) $ is a $ C^{\*} $-subalgebra of $ A \times B $ also. Then as $ I\_{A} $ is the image of $ R \cap (A \times \{ 0\_{B} \}) $ under the projection $ \* $-homomorphism
$$
\pi: A \times B \to A,
$$
it follows that $ I\_{A} $ is a $ \* $-subalgebra of $ A $. To conclude that $ I\_{A} $ is a *closed* $ \* $-subalgebra of $ A $, which would make it a $ C^{\*} $-subalgebra of $ A $, use the well-known fact that any $ \* $-homomorphism from one $ C^{\*} $-algebra to another has a closed range.


It remains to show that $ I\_{A} $ is a two-sided ideal of $ A $. To begin, let $ a \in I\_{A} $ and $ x \in A $. By the hypotheses on $ R $, there exists a $ b \in B $ such that $ (x,b) \in R $. Then as $ R $ is a subalgebra of $ A \times B $, we obtain
$$
(a x,0\_{B}) = (a,0\_{B}) (x,b) \in R \quad \text{and} \quad
(x a,0\_{B}) = (x,b) (a,0\_{B}) \in R.
$$
Therefore, $ a x,x a \in I\_{A} $, and as $ x $ is arbitrary in $ A $, we deduce that $ I\_{A} $ is indeed a closed two-sided ideal of $ A $.


A similar argument works to show that $ I\_{B} $ is a closed two-sided ideal of $ B $. $ \quad \blacksquare $



> 
> **Claim 2:** $ A / I\_{A} $ and $ B / I\_{B} $ are isomorphic $ C^{\*} $-algebras.
> 
> 
> 


*Proof of Claim 2*


Define a map $ \Phi: A / I\_{A} \to B / I\_{B} $ by
$$
\forall a \in A: \quad
\Phi(a + I\_{A}) \stackrel{\text{df}}{=} b + I\_{B},
$$
where $ b $ is *any* element of $ B $ such that $ (a,b) \in R $ (such a $ b $ must exist). We now establish the following:


* **$ \Phi $ is well-defined.** Let $ (a,b),(a',b') \in R $, so that $ (a - a',b - b') \in R $. Furthermore, suppose that $ a - a' \in I\_{A} $, i.e., $ (a - a',0\_{B}) \in R $. Then
$$
(0\_{A},b - b') = (a - a',b - b') - (a - a',0\_{B}) \in R,
$$
so $ b - b' \in I\_{B} $.
* **$ \Phi $ is injective.** Let $ (a,b) \in R $, and suppose that $ b \in I\_{B} $, i.e., $ (0\_{A},b) \in R $. Then
$$
(a,0\_{B}) = (a,b) - (0\_{A},b) \in R,
$$
so $ a \in I\_{A} $.
* **$ \Phi $ is surjective.** Given any $ b \in B $, the requirements on $ R $ guarantee that there exists an $ a \in A $ such that $ (a,b) \in R $. Hence, $ \Phi(a + I\_{A}) = b + I\_{B} $.


This concludes the proof of Claim 2. $ \quad \blacksquare $



> 
> **Claim 3:** The required $ C^{\*} $-algebra $ C $ is $ B / I\_{B} $.
> 
> 
> 


*Proof of Claim 3*


Define a surjective $ \* $-homomorphism $ f: A \to B / I\_{B} $ by $ f \stackrel{\text{df}}{=} \Phi \circ q\_{A} $, where $ q\_{A} $ is the quotient map from $ A $ to $ A / I\_{A} $. Similarly, define a surjective $ \* $-homomorphism $ g: B \to B / I\_{B} $ simply by $ g \stackrel{\text{df}}{=} q\_{B} $, where $ q\_{B} $ is the quotient map from $ B $ to $ B / I\_{B} $.


Let $ (a,b) \in A \times B $.


* If $ (a,b) \in R $, then
$$
f(a) = \Phi({q\_{A}}(a)) = \Phi(a + I\_{A}) = b + I\_{B} = g(b).
$$
* Conversely, if $ f(a) = g(b) $, then by the same relations above, we have $ \Phi(a + I\_{A}) = b + I\_{B} $. From the definition of $ \Phi $, there exists a $ b' \in B $ satisfying $ (a,b') \in R $ such that $ b + I\_{B} = b' + I\_{B} $. Hence,
$$
 (a,b)
= (a,b') + \underbrace{(0\_{A},b - b')}\_{\text{$ \in R $, as $ b - b' \in I\_{B} $}} \in R,
$$
which concludes the proof of Claim 3. $ \quad \blacksquare $


