Let $A$ and $B$ be two unital infinite-dimensional simple separable nuclear $C^{\ast}$-algebras and let $C$ be a CAR-algebra. When does $A\otimes C \simeq B\otimes C$, imply $A\simeq B$? 


The answer is clear if $A$ and $B$ absorb the CAR-algebra $C$. Although, I don't know what type of algebras can absorb a CAR-algebra.



Tensoring  with a CAR-algebra
The CAR-algebra is isomorphic to the UHF-algebra $M\_{2^{\infty}}$, i.e. the infinite tensor products of $M\_2(\mathbb{C})$. This is explained in Example 1.2.6 in the book [Classification of Nuclear, Simple $C^\*$-algebras"](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Rordam&s5=Classification%20of%20Nuclear&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=1878882) by Rordam. As an infinite UHF-algebra it is what is called *strongly self-absorbing*. This is a definition of Toms and Winter, which can be found [here](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Toms%20AND%20Winter&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=7&mx-pid=2302521). In particular, it implies that $M\_{2^{\infty}} \otimes M\_{2^{\infty}} \cong M\_{2^{\infty}}$, i.e. it absorbs itself. Unfortunately, I don't know if there is a classification of $C^\*$-algebras that absorb some infinite UHF-algebra. For other strongly self-absorbing algebras, there are such results. For example, a separable, simple and nuclear $C^\*$-algebra $A$ is purely infinite if and only if $A \otimes \mathcal{O}\_{\infty} \cong A$, where $\mathcal{O}\_{\infty}$ is the infinite Cuntz algebra (Theorem 7.2.6 in Rordam's book). 


If $A$ satisfies $A \otimes \mathcal{O}\_{\infty} \otimes \mathbb{K} \cong A$ and similarly for $B$ there might be a chance that you can reduce your first question to one about $KK$-groups using the classification results by Phillips, in particular Theorem 4.1.3 in the paper [A classification theorem for nuclear purely infinite simple $C^\*$-algebras](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Phillips&s5=Classification&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=1745197).


