Finite-dimensional invariant subspace property and amenability for a class of Banach algebras

Abstract

Motivated by a result of Ky Fan in 1965, we establish a characterization of a left amenable F-algebra (which includes the group algebra and the Fourier algebra of a locally compact group and quantum group algebras, or more generally the predual algebra of a Hopf von Neumann algebra) in terms of a finite dimensional invariant subspace property. This is done by first revealing a fixed point property for the semigroup of norm one positive linear functionals in the algebra. Our result answers an open question posted in Tokyo in 1993 by the first author (see [25, Problem 5]). We also show that the left amenability of an ideal in an F-algebra may determine the left amenability of the algebra. 1 2 A. T.-M. LAU AND Y. ZHANG invertible linear mappings that preserve the quadratic form J( x) = x 2 + y 2 + z 2 − c 2 t 2 , x = (x, y, z, t) ∈ R 4 where the constant c is the speed of light. Quantity J represents the space time interval. It is a well known fact that for any Lorentz transformation T there is a three dimensional subspace V of R 4 which is T -invariant and positive (in the sense that T ( x) = x and J( x) ≥ 0 for all x ∈ V ). L. S. Pontrjagin [43], I. S. Iovihdov [20], M. G. Krein [22, 21] and M. A. Naimark [40, 41] investigated infinite-dimension versions of the above invariant subspace property, and Naimark finally established the following theorem in 1963 [40].

Fixed point properties for semigroups of nonlinear mappings on unbounded sets

“…However, 1 (Z) does not have the fpp for bounded closed convex sets [9]. In a recent remarkable paper of Lin [25], it was shown that 1 (Z) can be renormed to have the fpp. This answers in negative a long-standing open question of whether every Banach space with the fpp is necessarily reflexive.…”

Algebraic and analytic properties of semigroups related to fixed point properties of non-expansive mappings

“…S ×E → E is continuous, then S is jointly continuous on compact sets. The following result was proved in [57].…”

“…As we have known, P 1 (A) is indeed a metric topological semigroup with the product and topology inherited from A. The following was proved in [57]. A semitopological semigroup S is extremely left amenable if LUC(S) has a multiplicative left invariant mean.…”

“…(F E ): Every jointly continuous representation of S on a compact Hausdorff space C has a common fixed point in C. For an F-algebra A it is pleasing that the left amenability of A is equivalent to the extreme left amenability of P 1 (A) as the authors revealed in [57]. Remark 4.9.…”

The mathematical work of Anthony To-Ming Lau

“…Finite-dimensional invariant subspace property of Ky Fan and the Hahn-Banach extension property for amenable Banach algebras. Motivated by a result of Ky Fan in 1965, the authors of [33] establish a characterization of a left-amenable F -algebra (which includes the group algebra and the Fourier algebra of a locally compact group, the predual L 1 (G) of a locally compact quantum group G, or, more generally, the predual algebra of a Hopf-von Neumann algebra) in terms of a finite-dimensional invariant subspace property. This is done by first exhibiting a fixed-point property for the semigroup of norm-one positive linear functionals on the algebra.…”