Uniformly bounded representation

In mathematics, a uniformly bounded representation formula_1 of a locally compact group formula_2 on a Hilbert space formula_3 is a homomorphism into the bounded invertible operators which is continuous for the strong operator topology, and such that formula_4 is finite. In 1947 Béla Szőkefalvi-Nagy established that any uniformly bounded representation of the integers or the real numbers is unitarizable, i.e. conjugate by an invertible operator to a unitary representation. For the integers this gives a criterion for an invertible operator to be similar to a unitary operator: the operator norms of all the positive and negative powers must be uniformly bounded. The result on unitarizability of uniformly bounded representations was extended in 1950 by Dixmier, Day and Nakamura-Takeda to all locally compact amenable groups, following essentially the method of proof of Sz-Nagy. The result is known to fail for non-amenable groups such as SL(2,R) and the free group on two generators. conjectured that a locally compact group is amenable if and only if every uniformly bounded representation is unitarizable.
Statement.
Let "G" be a locally compact amenable group and let "T""g" be a homomorphism of "G" into "GL"("H"), the group of an invertible operators on a Hilbert space such that
Then there is a positive invertible operator "S" on "H" such that "S" "T""g" "S"−1 is unitary for every "g" in "G".
As a consequence, if "T" is an invertible operator with all its positive and negative powers uniformly bounded in operator norm, then "T" is conjugate by a positive invertible operator to a unitary.
Proof.
By assumption the continuous functions
generate a separable unital C* subalgebra "A" of the uniformly bounded continuous functions on "G". By construction the algebra is invariant under left translation. By amenability there is an invariant state φ on "A". It follows that
is a new inner product on "H" satisfying
where
So there is a positive invertible operator "P" such that
By construction
Let "S" be the unique positive square root of "P". Then
Applying "S"−1 to "x" and "y", it follows that
Since the operators
are invertible, it follows that they are unitary.
Examples of non-unitarizable representations.
SL(2,R).
The complentary series of irreducible unitary representations of SL(2,R) was introduced by . These representations can be realized on functions on the circle or on the real line: the Cayley transform provides the unitary equivalence between the two realizations.
In fact for 0 < σ < 1/2 and "f", "g" continuous functions on the circle define
where
Since the function "k"σ is integrable, this integral converges. In fact
where the norms are the usual L2 norms.
The functions
are orthogonal with
Since these quantities are positive, ("f","g")σ defines an inner product. The Hilbert space completion is denoted by "H"σ.
For "F", "G" continuous functions of compact support on R, define
Since, regarded as distributions, the Fourier transform of |"x"|2σ – 1 is Cσ|"t"|−2σ for some positive constant Cσ, the above expression can be rewritten:
Hence it is an inner product. Let "H"'σ denote its Hilbert space completion.
The Cayley transform gives rise to an operator "U":
"U" extends to an isometry of "H"σ onto "H" 'σ. Its adjoint is given by
The Cayley transform exchanges the actions by Möbius transformations of SU(1,1) on S1 and of SL(2, R) on R.
The operator "U" interwtines corresponding actions of SU(1,1) on "H"σ and SL(2,R) on "H" 'σ.
For "g" in SU(1,1) given by
with
In fact it is easy to check that the operator 
λ("g")"T"λ("g")−1 – "T" has finite rank, with range"V""g", the finite-dimensional space of functions supported on the set of vertices joining "g" to the origin. For on any function vanishing on this finite set, "T" and λ("g")"T"λ("g")−1 are equal; and they both leave invariant "V""g", on which they acts as contractions and adjoints of each other. Hence if "f" has finite support and norm 1,
For |z| < 1/√3, these representations are all similar to the regular representation λ. If on the other hand 1/√3 < |z| <1, then the operator
satisfies
where "f" in "H" is defined by
Thus, if "z" is not real, "D" has an eigenvalue which is not real. But then π"z" cannot be unitarizable, since otherwise "D" would be similar to a self-adjoint operator.
The Dixmier Problem.
Jacques Dixmier asked in 1950 whether amenable groups are characterized by unitarizability, i.e. the property that all their uniformly bounded representations are unitarizable. This problem remains open to this day.
An elementary induction argument shows that a subgroup of a unitarizable group remains unitarizable. Therefore, the von Neumann conjecture would have implied a positive answer to Dixmier's problem, had it been true. In any case, it follows that a counter-example to Dixmier's conjecture could only be a non-amenable group without free subgroups. In particular, Dixmier's conjecture is true for all linear groups by the Tits alternative.
A criterion due to Epstein and Monod shows that there are also non-unitarizable groups without free subgroups. In fact, even some Burnside groups are non-unitarizable, as shown by Monod and Ozawa.
Considerable progress has been made by Pisier who linked unitarizability to a notion of factorization length. This allowed him to solve a modified form of the Dixmier problem.
The potential gap between unitarizability and amenability can be further illustrated by the following open problems, all of which become elementary if "unitarizable" were replaced by "amenable":