engleski

On unital C(X)-algebras and C(X)-valued conditional expectations of finite index

Let $X$ be a compact Hausdorff space and let $A$ be a unital $C(X)$-algebra, where $C(X)$ is embedded as a unital $C^*$-subalgebra of the centre of $A$. We consider the problem of characterizing the existence of a conditional expectation $E: A \to C(X)$ of finite index in terms of the underlying $C^*$-bundle of $A$ over $X$. More precisely, we show that if $A$ admits a $C(X)$-valued conditional expectation of finite index, then $A$ is necessarily a continuous $C(X)$- algebra, and there exists a positive integer $N$ such that every fibre $A_x$ of $A$ is finite- dimensional, with $\dim A_x \leq N$. We also give some sufficient conditions on $A$ that ensure the existence of a $C(X)$-valued conditional expectation of finite index.

C(X)-algebra ; C*-bundle ; conditional expectation of finite index ; non-commutative branched covering

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