I am getting a bit confused on the notation used for tensor products, is we have the tensor product space $V\otimes V^*$ if $v\in V$ and $a \in V^*$ then is the following correct?
$$v \otimes a=\langle a,v\rangle=a(v)=j(v)(a)=v(a)$$ where $\langle\;,\;\rangle$ is a bilinear map defined such that $\langle a,v\rangle=a(v)$ and $j(v)\in V^{**}$ defined such that $j(v)(a)\equiv a(v)$. And the $v(a)$ is not technically correct, but is taken to mean $j(v)(a)$.
The first and the last equalities are (fundamentally) wrong, the rest is ok. $v\otimes a$ is an element of $V\otimes V^\ast$, whereas $a(v)$ is a scalar (element of the ground field $k$); also $v(a)$ doesn't mean anything. I think what you intend to consider is the fact that the bilinear map $f:V\otimes V^\ast\rightarrow k$, $f(v\otimes a)=a(v)$, which induces a map on the tensor product $V\otimes V^\ast\rightarrow k$ which is such that $v\otimes a\mapsto a(v)$.