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• W2105201214 abstract "intimately connected with duality theory (the notation is that of [1]). In both cases the middle algebra is the closure of L(G) in the dual of the first algebra and also the predual of the third algebra (at least when G is amenable in the second case). Furthermore, the third algebra is closely connected with the multiplier algebra of the first algebra. For abelian groups, compact or discrete, Varopoulos [11], [12] showed to great effect how the second triple could be obtained and studied by starting with the tensor product ^0(G) (8)y^0(^)» 7 the greatest crossnorm. An analogous construction starting this time with^0(G ) ®x te0(G)9 A the least cross-norm, would produce the first triple. On the other hand, at least for amenable groups, the triples in (1) can be considered as the extreme case/?=1, 2, respectively, of a family {A(G), cv{G), B(G)}, 1 ^ / ? ^ 2 , associated with /^-convolution operator theory, and obtained by starting with the tensor product L'(G) <g)y L (G),pj±, or<g0(G) ®y LG), p=. Indeed, Herz has shown that A{G) is a pointwise Banach algebra [6] while B(G), l < p ^ 2 , is both the multiplier algebra of A{G) and the Banach dual space of cv(G), G amenable. In these notes we outline a new approach to convolution operator theory, by starting with ^^G) ®a ^0(G)9 a a tensorial norm [5], rather than with L'{G) ®y L (G). A triple { f (G), &'(G), #(<7)} analogous to (1) is obtained. For //-convolution operator theory, a family of tensorial norms CLVQ is used. The two basic ideas are to exploit classical Banach space theory concerning L(iu)spaces, for example, forgetting about group structure, and then, when a group structure is imposed, to exploit standard ^ 0 ( ^ ) a n d L(G)-techniques because all the 'L^-theory' has been thrown into the norm aM," @default.
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• W2105201214 date "1974-11-01" @default.
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• W2105201214 title "$L^p$-convolution operators and tensor products of Banach spaces" @default.
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• W2105201214 doi "https://doi.org/10.1090/s0002-9904-1974-13642-6" @default.
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