Following situation: If $\mathcal{O}\_\infty$ is the Cuntz algebra in infinitely many generators and $\mathcal{K}$ the compact operators on a separable Hilbert space, let $v\in \mathcal{O}\_\infty\otimes \mathcal{K}$ be a partial isometry. 
Questions:


1. Just to make shure: $v(\mathcal{O}\_\infty\otimes \mathcal{K})v^\*$ is endowed with the multiplication $(vxv^\*)(vyv^\*)=vxv^\*vyv^\*$ and the involution $(vxv^\*)^\*=vx^\*v$, where $x,y\in \mathcal{O}\_\infty\otimes \mathcal{K}$, a $C^\*$-algebra, is it? Or is $v(\mathcal{O}\_\infty\otimes \mathcal{K})v^\*$ not necessarily closed ?
2. (in this case that $v(\mathcal{O}\_\infty\otimes \mathcal{K})v^\*$ is a $C^\*$-algebra ).. Is there an embedding $v(\mathcal{O}\_\infty\otimes \mathcal{K})v^\*\to \mathcal{O}\_\infty$?


(edit: the following is false, as pointed out in Gabors answer) There is at least a $\*$-isomorphism $v(\mathcal{O}\_\infty\otimes \mathcal{K})v^\*\to \mathcal{O}\_\infty\otimes \mathcal{K}$ given by $vnv^\*\mapsto n$ where $n \in \mathcal{O}\_\infty\otimes \mathcal{K}$. H


owever, I don't know enough about $\mathcal{O}\_\infty$ to answer 2 (so, 2. remains my main question). 


Thank you. 



Is there an embedding $v(\mathcal{O}_\infty\otimes \mathcal{K})v^*\to \mathcal{O}_\infty$?
The answers to both questions is Yes.


For the first one, recall that a partial isometry satisfies the equation $v=vv^\*v$ with $p=vv^\*$ being a projection (with same range as $v$ if viewed as an operator on a Hilbert space), which easily prompts 
$$v(\mathcal O\_\infty\otimes\mathcal K)v^\* = p(\mathcal O\_\infty\otimes\mathcal K)p = \{ x\in\mathcal O\_\infty\otimes\mathcal K \mid x=pxp \}.$$
A C\*-subalgebra of this form is called a corner. If you want to show that it is closed, then assume $x=\lim\_{n\to\infty} x\_n$ with $x\_n=px\_np$ for some $x, x\_n\in \mathcal O\_\infty\otimes\mathcal K$, and you see that then $x=pxp$ must hold by continuity of multiplication.


Note that this first part uses nothing about $\mathcal O\_\infty\otimes\mathcal K$ being the C\*-algebra in this context, unlike the second part.


For the second part, one needs the fact that $\mathcal O\_\infty\otimes\mathcal K$ is a simple and purely infinite C\*-algebra. See the papers titled "Simple C\*-algebras generated by isometries" and/or "K-theory for certain simple C\*-algebras" by Joachim Cuntz if you want to know more. Anyway, from basic results on simple and purely infinite C\*-algebras, it is possible to find another partial isometry $w\in\mathcal O\_\infty\otimes\mathcal K$ with $w^\*w=p$ and $ww^\* \in \mathcal O\_\infty\otimes e\_{11} \cong \mathcal O\_\infty$. Then you can check that the assignment $x\mapsto wxw^\*$ yields a well-defined embedding from $p(\mathcal O\_\infty\otimes\mathcal K)p$ into $\mathcal O\_\infty\otimes e\_{11} \cong \mathcal O\_\infty$.



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Note also that this is false! This map is not even well-defined as a linear map, as you can see when you insert any projection that is orthogonal to $p$.


