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I am working on exercise 7.G in the book “K-Theory and C*-Algebras” by Wegge-Olsen.

Let $A$ be some unital C*-algebra, $u$ a unitary in $M_n(A)$, and $u’$ a standard unitary (which is defined to be a continuous function from $\mathbb{R}$ to $\mathbb{C}$ satisfying some properties). In the exercise, it says

Then $u_1:= 1\otimes u$ and $u_2:=\operatorname{diag}(u’,1,\dots, 1)\otimes 1$ are commuting unitaries in $B:=M_n((SA)^\sim)$.

Here $M_n((SA)^\sim)=(SA)^\sim \otimes M_n$ where $M_n$ denotes the algebra of $n\times n$ matrices, $SA$ is the suspension of $A$ and $(SA)^\sim$ is $SA$ with a unit adjoined.

I cannot see how $u_1$ and $u_2$ are commuting. I suppose $$M_n((SA)^\sim)\simeq (SA)^\sim \otimes M_n \subset C(\mathbb{T})\otimes A\otimes M_n.$$ However, $u\in M_n(A)$ and $\operatorname{diag}(u’,1,\dots, 1)\in C(\mathbb{T})\otimes M_n$. That is to say, we should write $$u_1=1\otimes \left(\sum_{i,j} u_{i,j}\otimes e_{i,j}\right),$$ $$u_2=u’\otimes e_{1,1}\otimes 1 + \sum_{k\not =1} 1\otimes e_{k,k}\otimes 1.$$ But it does not appear to me that $u_1$ and $u_2$ are commuting?

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  • $\begingroup$ I'm confused at the example for a standard unitary, $t\mapsto \exp(\frac{it}{1+|t|})$ is not in $C_0(\mathbb{R})^\sim$ since the limit dose not exist when $|t|\to +\infty$. Where is my problem? $\endgroup$ – C.Ding Nov 18 '18 at 1:27
  • $\begingroup$ I think $SA=\{f\in C([0,1]\to A): f(0)=f(1)=0\}=\{f\in C(\mathbb{T}\to A): f(1)=0\}$. $\endgroup$ – C.Ding Nov 18 '18 at 1:31
  • $\begingroup$ @C.Ding yes you’re right $\endgroup$ – Fan Nov 18 '18 at 5:13
  • $\begingroup$ @C.Ding I can’t see why the example of standard unitary works either. I was identifying $C_0(\mathbb{R})^\sim$ with $C(\mathbb{T})$ $\endgroup$ – Fan Nov 18 '18 at 6:17

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