Bimeasure algebras on locally compact groups (Q1081059)

For locally compact groups G, H let \(BM=BM(G,H)\) denote the Banach space of bounded bilinear forms on \(C_ 0(G)\times C_ 0(H)\), that is, the dual of the projective tensor product \(C_ 0(G)\otimes_{\gamma} C_ 0(H)\). In general, BM(G,H) does not arise as the completion of M(G)\(\otimes M(H)\) with respect to any tensorial norm. Elements \(u\in BM\) are called bimeasures. There is a fundamental result of Grothendieck which associates to each u regular Borel probability measures \(\mu\) on G, \(\nu\) on H such that \[ | u(f,g)| \leq K \| u\| \| f\|_{L^ 2(\mu)}\cdot \| g\|_{L^ 2(\nu)}\text{ for all } f\in C_ 0(G),\quad g\in C_ 0(H), \] where K is a universal constant. [In fact, G and H need only be locally compact spaces for this.] Using this the authors construct a multiplication and an adjoint operation in BM which generalize the convolution structure in M(G\(\otimes H)\) [i.e., the injection of the latter into the former becomes a *-algebra isomorphism] and make BM a Banach *-algebra, except that the constant \(K^ 2\) intervenes in the submultiplicativity of the norm. This algebra is then studied. The case of abelian G and H was investigated earlier by \textit{C. C. Graham} and the third author [Pac. J. Math. 115, 91-127 (1984; Zbl 0502.43005)] and the convolution and involution defined here agree in this case with those defined in that earlier paper (via Fourier transforms). A proof promised in this earlier work is now presented, in the non-abelian framework. It concerns a concept of continuous bimeasure u: sup\(\{| u(f,g)|:\) \(f\in C_ 0(U)\), \(g\in C_ 0(V)\}\to 0\) as open sets \(U\downarrow x\), \(V\downarrow y\), for every \(x\in G\), \(y\in H\). There is also a notion of discrete bimeasure. These two classes, \(BM_ c\), \(BM_ d\) are closed vector subspaces of BM, and BM is their topological direct sum [a result due to \textit{S. Saeki}, J. Math. Soc. Japan 28, 33-47 (1976; Zbl 0313.46022)]. Denoting the projection by the appropriate subscript, it is shown that \((u*v)_ d=u_ d*v_ d\) and from this follows easily that \(BM_ c\) is an ideal. The closure of \(L^ 1(G\times H)\) in BM is denoted by \(BM_ a=BM_ a(G,H)\); in fact this is \(L^ 1(G)\otimes_{\gamma} L^ 1(H)\). It is a closed ideal and \(u\in BM\) lies in \(BM_ a\) iff a certain translation operation \((x,y)\to R^*_{(x,y)}u\) is norm-continuous, analogous to the well-known characterization of \(M_ a(G)\) in M(G) (Plessner-Raikov theorem) [cf. \textit{D. A. Raikov}, Tr. Mat. Inst. Steklova 14 (1945; Zbl 0061.254)]. Moreover, the bounded linear operators T on \(BM_ a\) which satisfy \(T(u*v)=u*Tv\) [right mulipliers] are characterized, analogously to the \(M_ a(G)\) case, as right convolution by fixed elements of \(BM_ a\). However, there need not be a bounded projection of BM onto \(BM_ a\) and so no natural analog of singular measures need exist. It is shown that the natural map of \(BM(G_ 1,G_ 2)\to BM(G_ 1/H_ 1,G_ 2/H_ 2)\) is a surjective *-algebra homomorphism of norm 1 when \(G_ j\) are locally compact groups and \(H_ j\) are closed normal subgroups of them. Next, the lifting of unitary representations of G, H on Hilbert spaces \({\mathcal H}_ 1\), \({\mathcal H}_ 2\), respectively, to a *-representation of BM on \({\mathcal H}_ 1\otimes {\mathcal H}_ 2\) is studied. This is always uniquely realizable. Moreover, for type I groups it leads to the bounded inclusion relation \(BM_ a\subset C^*(G\times H)\) (the group \(C^*\)- algebra) and BM\(\subset VN(G\times H)\) (the group von Neumann algebra). Finally, returning to the abelian case, the authors find a natural injection of maximal ideal spaces : \(i: {\mathcal M}_{M(G)}\times {\mathcal M}_{M(H)}\to {\mathcal M}_{BM}\). This map is sometimes surjective [e.g., if either G or H is discrete], but not always. The authors note that for \(\pi\) the restriction map of \({\mathcal M}_{BM}\to {\mathcal M}_{M(G\times H)}\), the composite \(\pi\) i injects \({\mathcal M}_{M(G)}\times {\mathcal M}_{M(H)}\) into \({\mathcal M}_{M(G\times H)}\). The construction of i is based on the extension of elements of BM to bilinear functionals on \(C_ 0(G)^{**}\times C_ 0(H)^{**}.\) The paper is very clearly written. The authors provide good motivation and repeatedly remind the reader of the pervasive analogies with M(G). For example, there are two equivalent ways to define convolution in M(G) [\(\int_{G}f d(\mu *\nu):=\int_{G\times G}f(xy) d(\mu \times \nu)(x,y)\) and \(\int_{G}f d(\mu *\nu):=\int_{G}({\check \mu}*f) d\nu\) where \({\check \mu}\)*f(x):\(=\int_{G}f(xy) d\mu (x)]\) and there are two analogous and equivalent ways to do it in BM.