Homomorphisms from \(L^ 1(G)\) for \(G\in[FIA]^ -\cup[\text{Moore}]\)
One of the main problems in automatic continuity is the question of whether for a given Banach algebra \(A\) every homomorphism from \(A\) into an arbitrary Banach algebra \(B\) is continuous. Much is known in case \(A\) is a \(C^*\)-algebra. Although many \(C^*\)-algebras admit discontinuous homomorphisms, due to work of Sinclair and others there is a fair amount of information about the structure of homomorphisms. In the paper under review such problems are investigated for \(L^ 1\)- algebras \(L^ 1(G)\) of a locally compact group \(G\). Of course, one cannot expect substantial progress for general \(G\). The author obtains a number of interesting results for two classes of locally compact groups. The first one is [Moore], the class of groups all of whose irreducible unitary representations are finite dimensional, while the second, \([FIA]^ -\), consists of all groups \(G\) such that the group of inner automorphisms of \(G\) is relatively compact in the full automorphism group of \(G\), endowed with Birkhoff's topology. Both of these classes contain all abelian and all compact groups, and their structure and representation theory is well understood. If \(\Theta : A \to B\) is a homomorphism between Banach algebras, then the so-called continuity ideal \(I(\Theta)\) of \(\Theta\) is defined to be the set of all \(a \in A\) such that the mappings \(x \to \Theta(ax)\) and \(x \to \Theta(xa)\) from \(A\) into \(B\) are continuous. Using results from spectral synthesis it is shown that for \(G \in [\text{Moore}]\), \(\overline{I(\Theta)}\) has finite codimension in \(L^ 1(G)\). As a consequence, \(\Theta\) is continuous if and only if \(I(\Theta)\) is closed in \(L^ 1(G)\). Possibly, \(\Theta\) need not be continuous on \(I(\Theta)\). However, there always exists an ideal \(I\) in \(L^ 1(G)\) such that \(\overline{I}\) has finite codimension and the restriction \(\Theta| I\) is continuous. Turning to \(G \in [FIA]^ -\) and assuming that \(\overline{I(\Theta)}\) has finite codimension in \(L^ 1(G)\), similar results are obtained. Moreover, \(\Theta\) can be written as \(\Theta = \Theta_{\text{cont}} + \Theta_{\text{sing}}\) where \(\Theta_{\text{cont}}\) is a continuous homomorphism from \(L^ 1(G)\) into \(B\) which agrees with \(\Theta\) on a dense subalgebra of \(L^ 1(G)\) and \(\Theta_{\text{sing}}\) has some distinguished properties. If, in addition to \(\overline{I(\Theta)}\) having finite codimension in \(L^ 1(G)\), \(\Theta(L^ 1(G))\) is dense in \(B\) and \(B\) is supposed to be semisimple, then \(\Theta\) can be shown to be continuous. This latter result has very recently been extended considerably by the author himself [Intertwining operators over \(L^ 1(G)\) for \(G \in [PG] \cap [SIN]\), Math. Z. (to appear)]. It is well known that if \(G\) is an infinite abelian group (and assuming the continuum hypothesis), there always exists a discontinuous homomorphism from \(L^ 1(G)\) into some Banach algebra. In view of this the author conjectures that, for general \(G\), there is a discontinuous homomorphism from \(L^ 1(G)\) into some Banach algebra if and only if, for some \(n \in \mathbb{N}\), \(G\) has infinitely many inequivalent, \(n\)- dimensional, irreducible unitary representations. By a quite sophisticated construction the author is able to establish the ``if part'' of this conjecture for groups which contain a subgroup \(H \in [FIA]^ -\) of finite index.
scientific article; zbMATH DE number 599793