Full hereditary C*-subalgebra

Let $A$ be a $C^*$ -algebra and $B\subseteq A$ a hereditary $C^*$ -subalgebra. The subalgebra $B$ is full in $A$ if it is contained in no proper closed two-sided ideal of $A$ . Equivalently, the ideal generated by $B$ is all of $A$ :

\overline{\operatorname{span}}\,ABA=A.

Fullness is therefore an ambient property of the inclusion $B\subseteq A$ ; the same $C^*$ -algebra embedded hereditarily in two different ambient algebras may be full in one and not in the other.

Full corners

If $p$ is a projection in $A$ , or suitably in its multiplier algebra, then $pAp$ is hereditary. It is a full corner exactly when $p$ is a full projection, meaning $\overline{\operatorname{span}}\,ApA=A$ . Every full corner is thus a full hereditary subalgebra, but a general hereditary subalgebra need not be presented by a projection belonging to $A$ .

Morita-theoretic significance

A full hereditary $C^*$ -subalgebra $B$ is strongly Morita equivalent to $A$ , with the completion of $BA$ providing the equivalence module. Under $\sigma$ -unital hypotheses, the Brown–Green–Rieffel theorem further relates this equivalence to stable isomorphism.

Examples and non-examples

For a nonzero projection $p\in\mathcal K(H)$ , the corner $p\mathcal K(H)p$ is full because the compact-operator algebra is simple. By contrast, if $A=A_1\oplus A_2$ , then $A_1\oplus0$ is hereditary but not full: the ideal it generates is only $A_1\oplus0$ .

References

Marc A. Rieffel, “Morita Equivalence for $C^*$ -Algebras and $W^*$ -Algebras,” Journal of Pure and Applied Algebra 5 (1974), 51–96. DOI record. Relevant: §§1–2 on full hereditary subalgebras and equivalence bimodules.
Lawrence G. Brown, Philip Green, and Marc A. Rieffel, “Stable Isomorphism and Strong Morita Equivalence of $C^*$ -Algebras,” Pacific Journal of Mathematics 71 (1977), 349–363. Journal PDF. Relevant: Theorem 1.2 and the discussion of full hereditary subalgebras.