extensions of states on tensor products

Question

Let $A, B$ two $C^*$-algebras, $J$ an ideal in $B$ (two-sided, closed) and denote with $A\odot J$ the *-algebraic tensor product of $A$ and $J$.

Some definitions: A linear map $f:A\odot J\to \mathbb{C}$ is called positive, if $f(x^*x)\ge 0$ for all $x\in A\odot J$. Furthermore, for positive $f:A\odot J\to \mathbb{C}$ we define $\|f\|=\sup\{f(a\otimes j): a\in A_+, \|a\|\le 1, j\in J_+, \|j\|\le 1\}$ (here $A_+$ denotes the positive elements of $A$). We define $S(A\otimes J)$ to be the set of all positive maps $f:A\odot J\to \mathbb{C}$ with $\|f\|=1$.

Fact (I'm not sure if it's important for my problem): $S(A\odot J)$ is homeomorphic to the state space $S(A\otimes_{max} J)$ (both sets are endowed with the weak*-topology), in symbols: $S(A\odot J)\cong S(A\otimes_{max} J)$.

Now I want to prove that for each $f\in S(A\odot J)\cong S(A\otimes_{max} J)$ there is an extension $\hat{f}\in S(A\odot B)\cong S(A\otimes_{max} B)$:

Let $f\in S(A\odot J)\cong S(A\otimes_{max} J)$ and $(u_t)_t\subseteq J$ an approximate unit. Then define $\hat{f}:A\odot B\to \mathbb{C}$ as follows: $$\hat{f}(a\otimes b):=\lim\limits_t f(a\otimes u_t bu_t).$$

My problem is now to prove that $\hat{f}$ is positve and $\|\hat{f}\|\le 1$. For the positivity, I'm not sure how to write $a\otimes u_t bu_t$ as a compression $s^*(a\otimes b)s$ for $s\in A\odot J$ (or $s\in A\otimes_{max} J$, respectively). Now my questions are (question 3. is not important for my main problem):

1. how to prove that $\hat{f}$ is positive?
2. why is $\|\hat{f}\|\le 1$?
3. Is such an extension $\hat{f}$ unique?

I appreciate your help.

Answer 1

The writing of your question hints that you are operating under the (wrong) assumption that the elements of $A\otimes B$ are always elementary tensors. The tensor product is an algebra, so it contains sum of elementary tensors.

Let $x=\sum_j a_j\otimes b_j$. Then \begin{align} \hat f(x^*x)&=\sum_{k,j}\hat f(a_k^*a_j\otimes b_k^*b_j) =\lim_t\sum_{k,j} f(a_k^*a_j\otimes u_tb_k^*b_ju_t)\\ \ \\ &=\lim_t f\left(\left(\sum_j a_j\otimes b_ju_t \right)^* \left(\sum_j a_j\otimes b_ju_t \right)\right)\geq0 \end{align} For the norm, let $\{v_r\}$ be an approximate unit for $A$. Then \begin{align} |\hat f(x)|&=\lim_t\left|f\left(\sum_j a_j\otimes u_tb_ju_t\right) \right|\\ \ \\ &=\lim_t\lim_r\left|f\left((v_r\otimes u_t)\left(\sum_j a_j\otimes b_j\right)\,(v_r\otimes u_t)\right) \right|\\ \ \\ &\leq\limsup_t\limsup_r\|f\|\,\|v_r\otimes u_t\|^2\,\left\|\sum_j a_j\otimes b_j\right\|\\ \ \\ &=\|f\|\,\|x\|=\|x\|. \end{align} Then $\|\hat f\|\leq1$.