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As far as I'm concerned $C^*$ bimodules generally denote those you attributed to A.Connes. Such a bi-module define (by tensorization over B) a functor from $B$ $C^*$-modules to $A$ $C^*$-modules. Any …

$\pi$ is an irreducible representation. So because of Schur lemma and the double commutant theorem the map $\pi:A \rightarrow B(H)$ is surjective. Now the state $\phi$ is defined as a normale state o …

Here is one half of an answer, but it was too long for a comment. I'll edit if I manage to finish it... When you have a $C^*$ algebra $C$ included in a von Neumann algebra $B$ ( and you might want to …

Because the group is compact one can assume the representation is isometric and the Haar measure is normalized. In this situation, $T$ is just the orthogonal projection on the space of $G$-invariant v …

Let assume that you consider unital algebra only (one can still study non unital algebra by unitarizing them, but notion of spectrum is always a little annoying when one want to consider non unital al …

Given two locally compact spaces $X$ and $Y$ then the product $X \times Y$ is an open subset inside $\overline{X} \times \overline{Y}$, where $\overline{X}$ and $\overline{X}$ are the one point compac …

Assume $A$ and $B$ are two unital $C^*$-algebras which are strongly Morita equivalent. This means that there exists a Hilbert $A$-module $H$ such that $B$ is the algebra of compact operators on $H$. …