Relations between the homologies of \(C^*\)-algebras and their commutative \(C^*\)-subalgebras (Q2777983)

Biprojective Banach algebras are among the homologically best algebras. A Banach algebra \(A\) is biprojective if it is projective as a Banach \(A\)-bimodule (or equivalently, the product morphism \(\pi:A\widehat{\otimes}A\rightarrow A\) has a right inverse in the category of \(A\)-bimodules) [cf. \textit{A. Ya. Khelemskij}, Math. USSR, Sb. 16, 125-138 (1972; Zbl 0245.46070)]. A Banach algebra \(A\) is called hereditarily projective if every closed left ideal of \(A\) is projective. NEWLINENEWLINENEWLINEThe author shows that if \(A\) is a \(C^*\)-algebra, \(B\) a commutative \(C^*\)-subalgebra of \(A\), \(I\) a closed ideal of \(B\), and the closed linear span \(\overline{AI}\) of \(\{as\); \(a\in A\), \(s\in I\}\) is projective as a left Banach \(A\)-module, then the left Banach \(B\)-module \(I\) is projective. As a corollary she establishes that if a \(C^*\)-algebra has the property that every closed left ideal is projective then the same is true for all its commutative \(C^*\)-subalgebras, and she concludes that no infinite dimensional \(AW^*\)-algebra is hereditarily projective. NEWLINENEWLINENEWLINEIt is shown that if \(A\) is a commutative \(C^*\)-algebra acting on a separable Hilbert space, then \(A\) is separable if and only if the spatial \(C^*\)-tensor product \(A\bigotimes_{\min}A\) is hereditarily projective, if and only if the spectrum of \(A\) is metrizable and has a countable base. NEWLINENEWLINENEWLINEThe author proves that every commutative \(C^*\)-algebra of a biprojective \(C^*\)-algebra has discrete spectrum and is biprojective. Moreover, she gives a new proof for the well-known result that any biprojective \(C^*\)-algebra is a direct sum of \(C^*\)-algebras of the type \(M_n(\mathbb C)\) [cf. \textit{Yu. V. Selivanov}, Funct. Anal. Appl. 10, 78-79 (1976); translation from Funkts. Anal. Prilozh. 10, No. 1, 89-90 (1976; Zbl 0327.46059)].