Homological properties of Banach and C*-algebras of continuous fields

Abstract

One concern in the homological theory of Banach algebras is the identification of projective algebras and projective closed ideals of algebras. Besides being of independent interest, this question is closely connected to the continuous Hochschild cohomology. In this thesis we give necessary and sufficient conditions for the left projectivity and biprojectivity of Banach algebras defined by locally trivial continuous fields of Banach algebras. We identify projective C*-algebras A defined by locally trivial continuous fields U = fW, (At)t2W,Qg such that each C*-algebra At has a strictly positive element. We also identify projective Banach algebras A defined by locally trivial continuous fields U = fW, (K(Et))t2W,Qg such that each Banach space Et has an extended unconditional basis. In particular, for a left projective Banach algebra A defined by locally trivial continuous fields U = fW, (At)t2W,Qg we prove that W is paracompact. We also show that the biprojectivity of A implies that W is discrete. In the case U is a continuous field of elementary C*-algebras satisfying Fell’s condition (not nessecarily a locally trivial field) we show that the left projectivity of A defined by U, under some additional conditions on U, implies paracompactness of W. For the above Banach algebras A we give applications to the second continuous Hochschild cohomology group H2(A, X) of A and to the strong splittability of singular extensions of A.