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4

In short, the objective of the classification program is to give a (relatively) simple set of invariants that distinguish between the isomorphism classes of $C^\ast$-algebras, that is, if the invariants are isomorphic, then the $C^\ast$-algebras are isomorphic. It is thus stronger than what homology and cohomology theories yield for topological spaces: ...


3

To expand on my comment, let us take as given the following two (nontrivial) facts: Lemma 1: If $U$ is a $C^*$-algebra and $I\subset U$ is a closed 2-sided $*$-ideal, then $U/I$ is a $C^*$-algebra. Lemma 2: If $U$ and $V$ are $C^*$-algebras and $\pi:U\to V$ is an injective $*$-morphism, then $\|\pi(A)\|=\|A\|$ for all $A\in U$. Now suppose ...


2

As you wrote, you have that $s=\lim t_ns_n^*s_n$. You also have that $1=\lim s_n^*s_n$. Then $$ \|s-t_n\|\leq\|s-t_ns_n^*s_n\|+\|t_ns_n^*s_n-t_n\|=\|s-t_ns_n^*s_n\|+\|t_n(s_n^*s_n-1)\|. $$ The first term on the right goes to zero, and the second will if the sequence $\|t_n\|$ is bounded. The norms of the $t_n$ will stay bounded if the $s_n^*s_n$ are forced ...


1

A late answer, but maybe still helpful. All you have to keep in mind are the natural identification of $\ell^1$ and $c_0^\ast$ resp. $\ell^\infty$ and $(\ell^1)^\ast$. I will write $\ast$ for the product in the three steps of the construction of the Arens product so that there is no confusion with pointwise multiplication. For $a,b\in c_0, \omega\in\ell^1$ ...


1

Since you have to relate the norm with the spectrum, I don't think this has an elementary algebraic proof. Facts needed (from the theory of Banach algebras): For any $A\in\mathcal S$, $\sigma(A)=\{\phi(A):\ \phi\in S(\mathcal A)\}$, where $S(\mathcal A)$ is the state space. For any $A\in\mathcal S$, $\|A\|=\text{spr}(A)=\max\{|\lambda|:\ ...


1

Yes and you don't even have to take the closure. For instance if $A$ is a non-simple C*-algebra with trivial cetnre that has unique (faithful) trace (for instance $A=C^*(G)$ for a sufficiently non-commutative amenable group such as the group of permutations of integers that move at most finitely many entries), then $A\otimes \mathcal{Z}$, where $\mathcal{Z}$ ...



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