Contractive projections on Banach algebras
If \(A\) is a Banach algebra, then a projection \(P\) on \(A\) is called a conditional expectation (or quasi-expectation) if it satisfies \[ P(b_1 a b_2 )= b_1 (Pa) b_2 \qquad (b_1, b_2 \in P(A), \quad a \in A). \] A classical theorem of J. Tomiyama asserts that, if \(A\) is a \(C^\ast\)-algebra and \(P\) is a contractive projection onto a \(C^\ast\)-subalgebra of \(A\), then \(P\) is automatically a conditional expectation. Consequently, the authors define a Banach algebra \(A\) to have the Tomiyama property if every contractive projection on \(A\) whose range is a subalgebra of \(A\) is a conditional expectation. Most of the paper is devoted to investigate, for a plethora of Banach algebras occurring naturally in abstract harmonic analysis, when those algebras have the Tomiyama property (or a variant of it). For instance, the authors prove that, for any locally compact group \(G\), the group algebra \(L^1(G)\) has the Tomiyama property for \(L\)-projections on \(^\ast\)-subalgebras, i.e., if \(P\) is projection on \(L^1(G)\) with \(\| f \| = \| P f \| + \| f - Pf \| \) for \(f \in L^1(G)\) with a \(^\ast\)-subalgebra as its range, then \(P\) is a conditional expectation.
scientific article; zbMATH DE number 5276379
scientific article