1
vote
1answer
35 views

Why is $K_{0}(C(\mathbb{D}))\rightarrow K_{0}(C(\mathbb{T}))$ injective?

There is the restriction map $\pi:C(\mathbb{D})\rightarrow C(\mathbb{T})$ where $\mathbb{D}$ is the closed unit disk and $\mathbb{T}$ is the unit circle. Why is $\pi_*:K_0(C(\mathbb{D}))\rightarrow ...
2
votes
0answers
31 views

If there is already enough room to add all projections, does passing to matrices change anything?

Throughout, $A$ denotes a $*$-algebra. We always assume $A$ is representable in the sense that $A$ can be embedded into $B(H)$ for some Hilbert space $H$. The particular embedding is not important, ...
1
vote
1answer
46 views

Can anyone give an example of two stably equivalent projections that are not Murray Von Neumann equivalent?

Two projections $P$, $Q$ are MvN equivalent in $C^*$-algebra $A$ when there is an element $u\in A$ such that $uu^*=P$ and $u^*u=Q$, and two projections $P$, $Q$ are stably equivalent if $P\oplus ...
1
vote
0answers
33 views

Strong and weak equivalence of $C^*$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
1
vote
1answer
100 views

Proving the inclusion map induces isomorphism on $K$-theory

Let $M$ be a $C^\ast$-algebra, $A, B$ be closed, two-sided ideals of $M$ such that $A+B=M$. Define $T=\{f\in C([0, 1], M):f(0) \in A, f(1) \in B\}$. Why is that the inclusion map of $C([0, 1], A\cap ...
2
votes
1answer
61 views

Connection in the KK-Theory

I have some questions about the connection in the KK-Theory. 1)The definition is complicated, why? What is the motivation? 2)Does any relation bewteen the connection at here with the differential ...
2
votes
0answers
75 views

Cap product between K-Theory and K-Homology

In Exercise 9.8.9 of the book "Analytic K-Homology" by Higson and Roe one has to construct a cap product $K_p(A) \otimes K^q(A) \to K^{q-p}(A)$, if A is commutative. Is the commutativity ...
4
votes
0answers
77 views

Products in $C^*$-algebra $K$-theory

Let $A_1$ and $A_2$ be unital $C^*$-algebras. If $p_1 \in M_{n_1}(A_1)$ and $p_2 \in M_{n_2}(A_2)$ are projections then $p_1 \otimes p_2 \in M_{n_1 n_2}(A_1 \otimes A_2)$ is also a projection, ...