Return to Question

Edit: According to the comment of Andre Henriques I revise the question:

What is an example of a noncommutative unital $C^\star$ algebra $A$, which is not Morita equivalent to a commutative algebra, such that for all unital subalgebra $B$ of $A$, $ K_{0}(B)$ has $\mathbb{Z}$ as a summand? This question is motivated by this post and the fact that commutative algebras and their matrix algebras satisfies the above property.

Edit: According to the comment of Andre Henriques I revise the question:

What is an example of a noncommutative unital $C^\star$ algebra $A$, which is not Morita equivalent to a commutative algebra, such that for all unital subalgebra $B$ of $A$, $ K_{0}(B)$ has $\mathbb{Z}$ as a summand? This question is motivated by this post and the fact that commutative algebras and their matrix algebras satisfies the above property.

What is an example of a noncommutative unital $C^\star$ algebra $A$ such that for all unital subalgebra $B$ of $A$, $ K_{0}(B)$ has $\mathbb{Z}$ as a summand? This question is motivated by this post and the fact that commutative algebras satisfies the above property.

Edit: According to the comment of Andre Henriques I revise the question:

What is an example of a noncommutative unital $C^\star$ algebra $A$, which is not Morita equivalent to a commutative algebra, such that for all unital subalgebra $B$ of $A$, $ K_{0}(B)$ has $\mathbb{Z}$ as a summand? This question is motivated by this post and the fact that commutative algebras and their matrix algebras satisfies the above property.