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On onesided harmonic analysis in non commutative locally compact groups. (English) Zbl 0399.22004

One-sided harmonic analysis on a locally compact group \(G\) means essentially the theory of one-sided ideals in the convolution algebra \(L^1(G)\) or more generally the theory of topological \(G\)-left-modules like \(L^\infty(G)\) or \(C_\infty(G)\), i.e. the algebra of continuous \(\mathbb C\)-valued functions on \(G\), vanishing at infinity or the Fourier-Eymard algebra \(A(G)\), which is a subalgebra of \(C_\infty(G)\).
Among the various notions of spectra for functions \(f\in L^\infty(G)\) for commutative groups \(G\) the most important is defined as the set \(\sigma(f)\) of all characters \(\chi\in \hat G\) the dual group, which are contained in the \(w^*\)-closed translation invariant subspace of \(L^\infty(G)\), generated by \(f\). The Wiener-Godement-Segal theorem states that for \(f\ne 0\) the spectrum \(\sigma(f)\) is never empty, or equivalently that every closed ideal \(I\subsetneq L(G)\) is annihilated by some non trivial unitary (= Hilbert space *-) representation.
For noncommutative groups this theorem in general is no longer true, thus the following problem was posed in [the author, Studia Math. 47, 37–49 (1973; Zbl. 258.22009)]: Determine the class \([w]\) of locally compact groups \(G\), for which every two-sided closed ideal \(I\ne L^1(G)\) is annihilated by some non trivial unitary representation of \(L^1(G)\). This is equivalent with the fact that every \(w^*\)-closed two-sided translation invariant subspace of \(L^\infty(G)\) contains non-zero positive definite functions.
The present paper originated from the investigation of the one-sided analogue of this problem: Under which conditions for \(G\) is it true that every non-zero left-translation invariant \(w^*\)-closed subspace of \(L^\infty(G)\) contains non-zero positive definite functions? For discrete \(G\) this is clearly equivalent with symmetry of \(L^1(G)\), but this is not the case in general:
One of the main results of the paper is the following theorem: A connected group \(G\) has this property if and only if \(G\) is the direct product of a compact group and an abelian group.
The principal tool to prove this theorem are the twisted algebras \(L^1(G, A)\) of \(A\)-valued integrable functions on \(G\), where \(A\) is an involutive translation invariant Banach subalgebra of \(C_\infty(X)\), with a homogeneous space \(X = G/K\) of cosets \(\dot x = xK\), \(K\) a compact subgroup of \(G\).
In parts I and II the case \(K= \{e\}\) is considered, i.e. \(X = G\), \(A\subset C_\infty(G)\). In this case \(\mathcal L = L^1(G,A)\) is topologically simple and symmetric, with the compact operators of a Hilbert space as its \(C^*\)-hull. \(A\) contains a Segal subalgebra \(A_1\), consisting of all \(u\in A\) with \(\int_G \vert u^x\bar u\vert_A\,dx < \infty\) with \(u^x(y) = u(xy)\). As the \(f\in\mathcal L\) can be considered as compact operators, the rank of \(f\) is well-defined. Then the rank one-elements in \(\mathcal L\) are precisely the \(f\in\mathcal L\) of the form \(f(x) = a^x\bar b\) with \(a, b\in A_1\). Moreover, there exists a natural lattice isomorphism between the lattices of closed left ideals in \(\mathcal L\) and of closed subspaces in \(A_1\), in particular one can classify those left ideals which are annihilated by positive definite linear functionals.
Similar, but more complicated results are obtained in part III for the more general case of a coset space modulo a compact subgroup \(K\) and if the finite rank elements are dense in \(\mathcal L = L^1(G,A)\). In this case \(\mathcal L\) is the closure of a direct sum of minimal two-sided simple ideals, each of which has the structure of the \(\mathcal L\)’s considered in part I.
In part IV the previous results are applied to motion algebras. These are algebras \(L^1(G, A)\) in which \(G\) is compact and \(A\) is some regular Tauberian Banach subalgebra of \(C_\infty(X)\) for a locally compact \(G\)-space \(X\).
Main results:
\(\mathcal L = L^1(G,A)\) is symmetric and Wiener, maximal two-sided ideals, primitive ideals and kernels of irreducible unitary representations are the same.
If \(G\) is connected then every closed left ideal is annihilated by positive definite linear forms if and only if \(G\) acts trivially on \(A\).
This is used in part V to prove eventually the theorem mentioned in the beginning. In a first step the proof is reduced to an essentially finite number of connected solvable Lie groups, resp. motion groups \(G\). For these \(G\) the algebras \(L^1(G)\) have quotients isomorphic to the twisted algebras studied in parts I to IV.

MSC:

22D15 Group algebras of locally compact groups
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A35 Positive definite functions on groups, semigroups, etc.
46H20 Structure, classification of topological algebras

Citations:

Zbl 0258.22009
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