On \(K_0\)-groups of operator algebras on Banach spaces (Q854648)

The first part of this paper is concerned with properties of indecomposable (\(I\) for short) and incompoundable (\(I_c\) for short) Banach spaces and related notions. A Banach space is indecomposable if it cannot be written as the topological direct sum of two infinite-dimensional closed subspaces and is called incompoundable if it cannot be written as the sum of two infinite-dimensional closed subspaces. One of the results can be expressed as \(Q.H.I=Q.H.I_c=H.I_c\) where, for a property \(P\) of Banach spaces, a space \(X\) has \(H.P\) if every closed, infinite-dimensional subspace of \(X\) has \(P\), and \(X\) has \(Q.H.P\) if every quotient of every subspace has \(P\). The second part studies some aspects of the \(K\)-theory of the algebra \(B(X)\) of all bounded linear operators on a Banach space \(X\). The authors examine the question when the \(K_0\)-group \(K_0(B(X))\) vanishes. They give an example to show that \(X\cong X\oplus X\) is not sufficient. They also present a result of Lin Chen: if \(X\) is isomorphic to its square and if, whenever \(X\) is written as a topological direct sum \(X=Y\oplus Z\), either \(Y\) or \(Z\) is isomorphic to its square, then \(K_0(B(X))=0\). Examples of such spaces are given, including direct sums of totally incomparable, prime Banach spaces.