T*-categories are introduced as a ternary generalization of C*-categories. Their linking C*-categories are constructed and the Gelfand-Naimark representation theorems of Zettl for C*-ternary rings and for W*-ternary rings, are generalized to T*-categories. Biduals of C*-categories and of T*-categories are considered.

Let A be a C*-algebra and ε: A → A a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, ε(x)*ε(x) ≤ ε(x*x), implies that ε(x)2 ≤ ε(x*x) In this note we show that ε is homomorphic (in the sense that ε(xy) = ε(x)E(y) for every x, y in A) if and only if ε(x)2 = ε(x*x), for every x in A. We also prove that a homomorphic conditional expectation on a commutative C*-algebra C0(X) is given by composition with a continuous retraction of X. One may therefore consider homomorphic conditional expectations as noncommutative retractions.

If B is a subalgebra of a von Neumann algebra A⊂B(H) and B contains the rank one projections corresponding to an orthonormal basis of H, then a linear B-bimodule projection P on A with range B is of the form P(x)=∑jpjxpjx∈B(H) for orthogonal projections pj in A which are diagonal with respect to the basis. An analogous result holds if A=B(H) and B is a weakly closed ternary ring of operators.