Invariant means on a class of von Neumann algebras

Abstract

For G a locally compact group with associated von Neumann algebra VN(G) we prove the existence of an invariant mean on VN(G). This mean is shown to be unique if and only if G is discrete. 1. Definitions and preliminaries. Throughout we follow in general the notation of [3]. Let G be a locally compact group with identity e. Let L 1(G) be the group algebra of G with respect to left Haar measure and let C *(G) be the enveloping C *-algebra of L 1(G). Denote by B(G) the dual space of C *(G). Then B(G) may be realized as the space of all finite linear combinations of continuous positive-definite functions on G. With the dual norm and pointwise multiplication, B(G) is a commutative Banach algebra-the Fourier-Stieltjes algebra of G. The closed subalgebra A(G) generated by elements with compact support is called the Fourier algebra of G. Let P = u e A(G): u is positive definite and llu|l = u(e) = 1}. For x a complex-valued (a.e. defined) function on G, denote by X, x the functions x(g) = x(g 1), x(g) = x(g) where denotes complex conjugation. If / C L1(G), denote by Lf the bounded operator on L2(G) defined by Lfx = / * x (where * denotes convolution). An important property of A(G) is that it is precisely the set of functions of the form x * y where x, y e L2(G) [3, p. 2181. Let VN(G) denote the vonNeumann algebra on L 2(G) generated by the operators Lf where / e L 1(G). Then VN(G) may be regarded as the dual space of A(G) under the map (T, u) (Tx, y) if u e A(G) with ui = y *X. In particular, the w*and weak operator topologies on VN(G) coincide. For u C A(G), T C VN(G) define the operator uT e VN(G) by (uT, v) = T, uv), v C A(G). It follows readily that VN(G) is an A(G)-module and that |luT|| < Ilull 1ITIh. A linear functional m on VN(G) is called a mean if Received by the editors June 28, 1971. AMS 1970 subject classfications. Primary 22D15, 43A15; Secondary 22D25, 43A07, 43A40.