I am working on my Masters thesis which is on the Grothendieck's inequality. My aim is to study its formulation in different contexts starting from commutative case and then covering non-commutative cases (C*-algebra and operator space versions). Currently, I am following the CBMS notes by Gilles Pisier reading select portions as per need.
While working through the C*-algebra case I am stuck at the following point. I have the following two theorems.
The theorem which can be called as non-commutative litte Grothendieck Theorem.
Theorem 9.4. Let $A$ be a C*-algebra, $H$ be a Hilbert space and let $T:A\rightarrow H$ be an operator. Then there are two states $f,g$ on $A$ such that for all $x\in A$ we have $$ \|Tx\|\leq \|T\|(f(x*x)+g(xx*))^{1/2}$$
The second theorem is about factorization of an operator.
Theorem 4.1. Let $X,Y$ be Banach spaces such that $X^*$ and $Y$ are of cotype 2. Then every approximable operator $u:X\rightarrow Y$ factors through a Hilbert space and satisfies $$\gamma_2(u)\leq (2C_2(X^*)C_2(Y))^{3/2}.$$
We recall that if $T:X\rightarrow Y$ factors through a Hilbert space $H$ and $T=AB$ where $B:X\rightarrow H $ and $A:H\rightarrow Y$ then we define $\gamma_2(u)=\inf \|A\|\|B\|$ where the infimum runs over all possible factorizations. Also $C_2(X^*)$ and $C_2(Y)$ are cotype constants of $X^*$ and $Y$ respectively.
Now, using these two theorems we have to prove the following theorem.
Theorem 9.6. Let $A$ be a C*-algebra and $Y$ be of cotype 2. Then any approximable operator $T:A\rightarrow Y$ factors through a Hilbert space. Moreover, there are states $f$ and $g$ on $A$ such that for all $x\in A$ we have $$\|Tx\|\leq C\|T\|(f(x*x)+g(xx*))^{1/2},$$ where $C$ is a constant depending only on the cotype 2 constant of Y.
I proceed for the proof in the following obvious way.
Proof. Since $A$ is a C*-algebra, then $A^*$ is of cotype 2. Then by theorem 4.1, $T$ factors through a Hilbert space, and $$\gamma_2(T)\leq (2C_2(A^*)C_2(Y))^{3/2}=C\quad \text{say}.$$ Let the factorization of $T$ be $T_1:A\rightarrow H$ and $T_2:H\rightarrow Y$ and $T=T_2T_1$. Then \begin{equation} \|Tx\|=\|T_2T_1(x)\leq \|T_2\|\|T_1(x)\|. \end{equation} Since $T_1:A\rightarrow H$ we apply theorem 9.4 to get states $f,g$ on $A$ such that \begin{equation} \|T_x\|\leq \|T_1\| (f(x*x)+g(xx*))^{1/2} \end{equation} so that \begin{equation} \|Tx\|\leq \|T_2\|\|T_1\| (f(x*x)+g(xx*))^{1/2} \end{equation}
Here is the point where I am stuck. How do I show that the RHS part in the inequality is $\leq C'\|T\|(f(x*x)+g(xx*))^{1/2}$, where $C'$ is come constant which depends only on the cotype 2 constant of $Y$.
Another technicality is that I don't want to go in the theory of type and cotype. Instead I am taking theorem 4.1 for granted and that the dual of a C*-algebra is of cotype 2. Rest I am just trying to merge the results well. My guide says that it has something to do with approximable property. But I am not sure of that. I would appreciate if one could show how precisely I should get the constant.
