# Commuting Elements in Tensor Products of C*-Algebras

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?

• 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? – C.Ding Nov 18 '18 at 1:27
• I think $SA=\{f\in C([0,1]\to A): f(0)=f(1)=0\}=\{f\in C(\mathbb{T}\to A): f(1)=0\}$. – C.Ding Nov 18 '18 at 1:31
• @C.Ding yes you’re right – Fan Nov 18 '18 at 5:13
• @C.Ding I can’t see why the example of standard unitary works either. I was identifying $C_0(\mathbb{R})^\sim$ with $C(\mathbb{T})$ – Fan Nov 18 '18 at 6:17