Support of a subalgebra of a $C^*$ algebra

Question

Let $A$ be a finite dimensional $C^*$ algebra and let $I$ be a two sided ideal in $A$. What is meant by the notion of support of $A/I$. I have heard of support of a self adjoint operators.Can any one please provide the reference for this?

Answer 1

Never heard of this, but it is not hard to imagine what it could be.

A finite-dimensional C$^*$-algebra is of the form $$ A=\bigoplus_{k=1}^m M_{n(k)}(\mathbb C). $$ Ideals are of the form $$ I=\bigoplus_{k=1}^m \,\alpha_k\,M_{n(k)}(\mathbb C), $$ where $\alpha_k\in\{0,1\}$. That is, the only way to obtain an ideal is to make some of the components equal to zero. Equivalently, $I=pA$ where $p=\bigoplus_{k=1}^m \alpha_k\,I_{n(I)}$ is a central projection.

It is easy to check that $A/I\simeq (1-p)A$, that is $$ A/I\simeq\bigoplus_{k=1}^m \,(1-\alpha_k)\,M_{n(k)}(\mathbb C). $$ So one could say that $1-p$ is the "support" of $A/I$.