Representation-Dependent Properties of $\mathrm{C}^*$-algebras Consider a $\mathrm{C}^*$-algebra. By the Gelfand-Naimark Theorem it has a faithful representation $A\subset B(\mathsf{H})$ of bounded operators on a Hilbert space. Let $\pi_1:A\rightarrow B(\mathsf{H}_1)$ and $\pi_2:A\rightarrow B(\mathsf{H}_2)$ be two faithful representations. These representations are called unitarily equivalent if there exists a unitary intertwiner $U:\mathsf{H}_1\rightarrow \mathsf{H}_2$, such that $U\pi_1=\pi_2U$.
Let us quote nLab:

From the physical viewpoint unitarily equivalent representations describe the same system, so that the classification of not unitarily equivalent representations is an important topic.

QUESTION 1

What properties of a $\mathrm{C}^*$-algebras are representation dependent? Do these properties coincide for unitarily equivalent faithful representations?

QUESTION 2

What is a good reference to read about unitary equivalence and such matters?


I will give an example of the kind of thing I am looking to hear about.
In this question we see that two projections can have trivial range intersection in one faithful representation and non-trivial in another. Note that this means that it is possible that if we look at $\pi_1(p),\,\pi_1(q)\in B(\mathsf{H}_1)$, and $\pi_2(p),\,\pi_2(q)\in B(\mathsf{H}_2)$, by von Neumann's theorem of alternating projections, both $\pi_1(pq)^n$ and $\pi_2(pq)^n$ converge strongly, but one to zero and the other to something non-zero. $\mathrm{C}^*$-algebras are not strongly closed, so this might be more about 'full' algebras bounded operators on Hilbert spaces... but the below I think gives something slightly more 'affiliated' with the $\mathrm{C}^*$-algebra
Consider the reduced group $\mathrm{C}^*$-algebra of the infinite dihedral group $D_\infty\cong \mathbb{Z}_2\star \mathbb{Z}_2$, denoted $\mathrm{C}^*_{\text{red}}(D_\infty)$. Faithfully represented on $\ell^2(D_\infty)$, where $u$ and $v$ are the free order two generators, the projections $p=(U_e+U_u)/2$ and $q=(U_e+U_v)/2$... I believe these have empty intersection (there is another faithful representation of this $\mathrm{C}^*$-algebra of Pederson in which I am more sure of this empty-range-intersection but harder to describe), but the counit, as $D_\infty$ is amenable, $\varepsilon(U_g)=1$ is a bounded linear functional, and if the GNS space of the counit is constructed, and included in a direct sum type faithful representation, then the cyclic vector of the counit is in the intersection of the ranges of $p$ and $q$. I do not know if these representations are unitarily equivalent (hence Question 2).
 A: This question is undoubtedly a very broad one so let me instead concentrate on a small contribution to the part related
to the intersection of the ranges of idempotent elements, which seems to be relevant for the OP.
So let us suppose throughout that $A$ is a C*-algebra and that  $p$ and $q$ are  projections in $A$.  Assuming that
there is a
representation $\pi $ such that the ranges of $\pi (p)$ and $\pi (q)$  have a nontrivial intersection then, picking a nonzero
vector $\xi $ in that intersection, we have that
$\pi (pq)\xi =\xi $,
which implies that $\Vert \pi (pq)\Vert =1$, and hence also  $\Vert pq\Vert =1$.
Conversely, if $\|pq\|=1$, then also
$$
  \|pqp\| = \|pq(pq)^*\| = \|pq\|^2 = 1.
  $$
Observing that $pqp$ is a positive element, we deduce that $1\in \sigma (pqp)$, so there exists a character $\varphi $ defined on the
C*-algebra generated by $pqp$, such that $\varphi (pqp)=1$.   Extending $\varphi $ to a state on $A$, and letting $\pi $ be the
associated  GNS representation, one has that
$$
  1 = \varphi (pqp) = \langle \pi (pqp)\xi ,\xi \rangle \leq \|\pi (pqp)\xi \Vert \Vert \xi \Vert  \leq 1.
  $$
From this one easily proves that $\pi (pqp)\xi =\xi $, whence $\xi $ lies in the range of $\pi (p)$.  Consequently
$$
  \|\xi \| = \|\pi (pqp)\xi \|=\|\pi (pq)\xi \|\leq \|\pi (q)\xi \|,
  $$
and we then deduce that $\xi $ lies in the range of $\pi (q)$ as well.
We have then proved:
Theorem.  Given two projections $p$ and $q$ in a C*-algebra $A$, there exists a representation $\pi $ of $A$ such that
the
ranges of $\pi (p)$ and $\pi (q)$  have a nontrivial intersection iff $\Vert pq\Vert =1$.

Speaking about the regular representation of ${\mathbb Z}_2\star{\mathbb Z}_2$, it is correct to say that the projections $p$ and $q$ defined in
the question have trivially intersecting ranges.  To see this, let me denote the images of the two standard generators
simply by $u$ and $v$,  so that
$$
  p = \frac{1+u}2\quad\text{and}\quad q = \frac{1+v}2.
  $$
If $\xi $ is a vector lying in the intersection of the ranges of $p$ and $q$,  then
$$
  \xi =p\xi  = \frac{\xi +u\xi }2,
  $$
from where one deduces that $u\xi =\xi $, and similarly that $v\xi =\xi $.  This would imply that $\xi $ is fixed for the whole of
${\mathbb Z}_2\star{\mathbb Z}_2$.
However  it is easy to see that the regular representation of an infinite group cannot have a fixed point
except for the zero vector, so we conclude that $p$ and $q$ indeed have trivially intersecting ranges.
On the other hand,  as observed by the OP,  the co-unit maps $pq$ to 1,  so $\|pq\|=1$,  and hence the above Theorem
could be used to find another representation (as the one constructed by the OP) in which the ranges have a non-trivial intersection.
