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Motivating by the concept in the following post What are these compact sets called? We introduce the following concept:

Let $A$ be a unital $C^{*}$ algebra. We consider the unitary equivalent relation on the set of all projections in $A$. We denote by $n(A)$, the number of equivalents classes of projections in $A$. The number of connected components of $A$ is denoted by $ncc(A)$ and is defined by $ncc(A)=log_{2} n(A)$. For a non unital $C^{*}$ algebra $A$, we define $ncc(A)=ncc(M(A))$ where $M(A)$ is the multiplier algebra of $A$. This is a natural definition because for a locally compact Hausdorff space $X$ we have $ncc(C_{0}(X))=$ The number of connected components of $X$.

The concept in the above post is a motivation to ask:

Question 1: Assume that We have an extension $0\to A\to B\to C\to 0$ of $C^{*}$ algebras with $ncc(A)< \infty$ and $ncc(C)< \infty$. Does this implies that $ncc(B)$ is finite, too? How can $ncc(C)$ be controlled by $ncc(A)$ and $ncc(C)$?

Question 2: How can $ncc(A\otimes_{min} B)$ be controlled in term of $ncc(A)$ and $ncc(B)$? When $A$ is commutative, is it true to say that $ncc(A\otimes B) \leq ncc(A).ncc(B)$?

Note: The commutative case gives us enough motivation for such noncommutative questions.

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As a sign of good faith, why don't you tell us what you've been able to work out when A is a noncommutative finite-dimensional Cstar algebra? –  Yemon Choi May 27 at 4:11

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