I am currently taking lectures on $C^*$-algebras. I have a question related to this topic, precisely on linear functional on these spaces.
Suppose $A$ is a $C^*$-algebra, with a faithful normal state $\phi$.
On $A^n$, we define an inner product by taking for $x,y\in A^n$, $\langle x,y\rangle_{\phi}=\sum_{k=1}^n\phi(y_i^*x_i)$,
My question is the following. If $T$ is a complex matrix in $M_n(\mathbb{C})$.
The action of $T$ on $A^n$ is define as usual by matrix multiplication over vectors.
Does the following inequality holds : $\lvert \langle Tx,y\rangle_{\phi}\rvert\leq\lVert T\rVert_{op}\lVert x\rVert\lVert y\rVert$.
With $op$ is the usual operator norm for matrices.
Thanks for you help.
#Edit# The question as been answered.
Yes.
You have, using first Cauchy-Schwarz for $\phi$ and then for the usual inner product in $\mathbb C^n$, \begin{align} |\langle Tx,y\rangle| &\leq\sum_k|\phi(y_k^*(Tx)_k)| \leq \sum_k\phi(y_k^*y_k)^{1/2}\phi((Tx)_k^*(Tx)_k)^{1/2}\\[0.3cm] &\leq\Big[\sum_k\phi(y_k^*y_k)\Big]^{1/2}\Big[\sum_k\phi((Tx)_k^*(Tx)_k)\Big]^{1/2}\\[0.3cm] &=\|y\|\,\Big[\sum_{k,j}\phi(x_j^*T_{kj}^*T_{kj}x_j)\Big]^{1/2}\\[0.3cm] &=\|y\|\,\Big[\sum_{k,j}\phi(x_j^*(T^*)_{jk}T_{kj}x_j)\Big]^{1/2}\\[0.3cm] &=\|y\|\,\Big[\sum_{j}\phi(x_j^*(T^*T)_{jj}x_j)\Big]^{1/2}\\[0.3cm] &\leq\|T\|\,\|y\|\,\Big[\sum_{j}\phi(x_j^*x_j)\Big]^{1/2}\\[0.3cm] &=\|T\|\,\|x\|\,\|y\|. \end{align}