Dgdrxrt

Context: Let $A$ be an ungraded (not necessarily unital) $C^*$ algebra. $mathcalK$ space of compact bounded operators on an infinite separable graded Hilbert space $H=H_0 oplus H_1$.
Consider the space
$$ A otimes mathcalK $$
Let us suppose there is a unique norm.

Edit: I replaced a large part of text which can be seen in history. For streamlining the post.

Claim 1' If we begin with a graded homomoprhism, $mathcalS rightarrow A otimes mathcalK$, then the unitary $u$ we obtain this way (via the Cayley transform) has the property that $alpha(u)=u^*$.

Claim 2: For any unital graded $C^*$ algebra $B$ containing $A otimes mathcalK$, consider the grading element,
$$
epsilon =
beginpmatrix
1 & 0 \
0 & -1
endpmatrix
$$
which grades $mathcalK$. Any skew unitary $u$ is equal to
$$p_epsilon = beginpmatrix
1 & 0 \
0 & 0
endpmatrix
$$
modulo $A otimes mathcalK$, i.e. $p_phi-p_epsilon in A otimes mathcalK$.

May someone elaborate the details? These are from page 43, proof of Prop 3.17 , Higson's notes.

Questions regarding Aweygan's reply

I am still confused:

1. Why is it to say that $ bin B$ lies in $A otimes mathcalK$ precisely means that the scalar part of each $(b_ij)$ is zero. (This is more or less related to Q3) and pehraps Exercise 3.1 in page 37 of notes. I would like to understand a related question.




2. $phi(f)$ is not a $2 times 2$ matrix here (?) i.e. why
   $$ phi(f) = (v_ij) in A otimes mathcalK?$$




3. Elaboration on 2 and how Aweygan did this: how we define $tildephi$: first decompose a function $C(S^1)$ as $c+f$, where $f in C_0(Bbb R)$, $c$ is a constant. define
   $$ tildephi(c+f) = cI + phi(f) $$
   we embed $phi(f) in A otimes mathcalK rightarrow B$. But what is this embedding?

For claim 1, the information copied here is not quite what they have stated in the book.

Suppose $A$ is a graded unital $C^*$-algebra, with the grading given by a $*$-automorphism $alpha:Ato A$. A unitary $uin A$ is called a skew-unitary if $alpha(u)=u^*$.

If the grading is internal, i.e., there is some self-adjoint unitary $varepsilonin A$ such that $alpha(x)=varepsilon xvarepsilon$ for all $xin A$, then the map from skew unitaries to projections given by $umapsto frac12(1+uvarepsilon)$ is a bijection.

With this information, it should be clear why $frac12(1+uvarepsilon)$ is a projection if $varepsilon$ is a self-adjoint unitary and $varepsilon uvarepsilon=u^*$.

For claim 2, the authors aren't claiming that any skew unitary is equivalent $p_epsilon$, only a very special one. In this section, $phi:mathcal Sto Aotimesmathcal K$ is a graded $*$-homomorphism. Via the Cayley transform, $phi$ induces a unital $*$-homomorphism $tildephi$ from $C(S^1)$ to the unitization of $Aotimesmathcal K$. The unitary in question is then $tildephi(z)$, where $z:S^1tomathbb C$ is the inclusion map.

Using the grading on $mathcal K$, we can consider the algebra $B$ in question as the algebra of all $2times 2$-matrices with entries in $widetildeAotimesmathcal K$ (the unitization of $Aotimesmathcal K$), graded by diagonal matrices (even part) and off-diagonal matrices (odd part). Then to say that $b=(b_ij)in B$ lies in $Aotimesmathcal K$ precisely means that the scalar part of each entry $b_ij$ is zero.

From the graded $*$-homomorphism $phi:C_0(mathbb R)to Aotimesmathcal K$, we obtain a unital $*$-homomorphism $tildephi:widetildeC_0(mathbb R)to B$ (by mapping units to units, and everything else by $phi$). Note that $widetildeC_0(mathbb R)=C(S^1)$ is generated by a single unitary $u$. Then $u=1+f$ for some $fin C_0(mathbb R)$, and
$$tildephi(u)=beginpmatrix1+v_11&v_12\v_21&1+v_22 endpmatrix$$
where $phi(f)=(v_ij)in Aotimesmathcal K$. Then we have
$$p_phi=frac12(1+tildephi(u)epsilon=frac12left(beginpmatrix1&0\0&1endpmatrix+beginpmatrix1+v_11&v_12\v_21&1+v_22 endpmatrixbeginpmatrix1&0\0&-1endpmatrixright)=beginpmatrix1+fracv_112&frac-v_122\fracv_212&frac-v_222endpmatrix,$$
so that
$$p_phi-p_epsilon=beginpmatrixfracv_112&frac-v_122\fracv_212&frac-v_222endpmatrixin Aotimesmathcal K.$$

Further Questions:

1. Note that $Aotimesmathcal K$ is isomorphic to $M_2(Aotimesmathcal K)$, by decomposing the Hilbert space $H$ that $mathcal K$ acts on into a direct sum $H=H_0oplus H_1$ (This is also how the grading on $mathcal K$ is defined). So when I say an element $(b_ij)$ of $B$ lies in $Aotimesmathcal K$ when the scalar part of each $b_ij$ is zero, I really mean that $(b_ij)$ lies in $M_2(Aotimesmathcal K)$.




2. As I said above, $Aotimesmathcal K$ is graded so that it looks like $M_2(Aotimesmathcal K)$. Thus the homomorphism $phi:mathcal Sto Aotimesmathcal K$ looks like a homomorphism $mathcal Sto M_2(Aotimesmathcal K)$.




3. How I have $B$ defined, a typical element of $B$ looks like a $2times 2$ matrix $(b_ij)=(a_ij+lambda_ij)$, where $a_ijin Aotimesmathcal K$ and $lambda_ijinmathbb C$. The embedding $Aotimesmathcal Kto B$ is just $(a_ij)mapsto(a_ij+0)$, with scalar part $0$.