Operator space structure and amenability for Figà-Talamanca-Herz algebras

Column and row operator spaces—which we denote by COL and ROW, respectively—over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p,p′∈(1,∞) with , we use the operator space structure on to equip the Figà-Talamanca-Herz algebra A_p(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p?q?2 or 2?q?p and amenable G, the canonical inclusion A_q(G)⊂A_p(G) is completely bounded (with cb-norm at most , where is Grothendieck's constant). As an application, we show that G is amenable if and only if A_p(G) is operator amenable for all—and equivalently for one—p∈(1,∞); this extends a theorem by Ruan.     

All-derivable points of operator algebras

Let A be an operator subalgebra in B(H), where H is a Hilbert space. We say that an element Z∈A is an all-derivable point of A for the norm-topology (strongly operator topology, etc.) if, every norm-topology (strongly operator topology, etc.) continuous derivable linear mapping φ at Z (i.e. φ(ST)=φ(S)T+Sφ(T) for any S,T∈A with ST=Z) is a derivation. In this paper, we show that every invertible operator in the nest algebra is an all-derivable point of the nest algebra for the strongly operator topology. We also prove that every nonzero element of the algebra of all 2×2 upper triangular matrixes is an all-derivable point of the algebra.     

Generalized biderivations of nest algebras

Let N be a nest on a complex separable Hilbert space H, and τ(N) be the associated nest algebra. In this paper, we prove that every biderivation of τ(N) is an inner biderivation if and only if dim 0₊ ≠ 1 or , and that every generalized biderivation of τ(N) is an inner generalized biderivation if dim 0₊ ≠ 1 and .     

A Morita type equivalence for dual operator algebras

We generalize the main theorem of Rieffel for Morita equivalence of W^∗-algebras to the case of unital dual operator algebras: two unital dual operator algebras A,B have completely isometric normal representations α,β such that α(A)=[M^∗β(B)M]^−w∗ and β(B)=[Mα(A)M^∗]^−w∗ for a ternary ring of operators M (i.e. a linear space M such that MM^∗M⊂M) if and only if there exists an equivalence functor which “extends” to a ∗-functor implementing an equivalence between the categories and . By we denote the category of normal representations of A and by the category with the same objects as and Δ(A)-module maps as morphisms (Δ(A)=A∩A^∗). We prove that this functor is equivalent to a functor “generated” by a B,A bimodule, and that it is normal and completely isometric.     

Crossed-products by finite index endomorphisms and KMS states

Given a unital -algebra A, an injective endomorphism preserving the unit, and a conditional expectation E from A to the range of α we consider the crossed-product of A by α relative to the transfer operator L=α⁻¹E. When E is of index-finite type we show that there exists a conditional expectation G from the crossed-product to A which is unique under certain hypothesis. We define a “gauge action” on the crossed-product algebra in terms of a central positive element h and study its KMS states. The main result is: if h>1 and E(ab)=E(ba) for all a,b∈A (e.g. when A is commutative) then the KMS_β states are precisely those of the form ψ=φ°G, where φ is a trace on A satisfying the identity

     

On Fourier algebra homomorphisms

Let G be a locally compact group and let B(G) be the dual space of C∗(G), the group C∗ algebra of G. The Fourier algebra A(G) is the closed ideal of B(G) generated by elements with compact support. The Fourier algebras have a natural operator space structure as preduals of von Neumann algebras. Given a completely bounded algebra homomorphism we show that it can be described, in terms of a piecewise affine map with Y in the coset ring of H, as follows

     

On some subalgebras of von Neumann algebras with analyticity

Let (M,α,G) be a covariant system on a locally compact Abelian group G with the totally ordered dual group which admits the positive semigroup . Let H^∞(α) be the associated analytic subalgebra of M; i.e. . Let be the analytic crossed product determined by a covariant system . We give the necessary and sufficient condition that an analytic subalgebra H^∞(α) is isomorphic to an analytic crossed product related to Landstad's theorem. We also investigate the structure of σ-weakly closed subalgebra of a continuous crossed product N?_θR which contains N?_θR₊. We show that there exists a proper σ-weakly closed subalgebra of N?_θR which contains N?_θR₊ and is not an analytic crossed product. Moreover we give an example that an analytic subalgebra is not a continuous analytic crossed product using the continuous decomposition of a factor of type III_λ(0?λ<1).     

On the stable rank of algebras of operator fields over metric spaces

Let Γ be a finitely generated, torsion-free, two-step nilpotent group. Let C^*(Γ) denote the universal C^*-algebra of Γ. We show that , where for a unital C^*-algebra A, sr(A) is the stable rank of A, and where is the space of one-dimensional representations of Γ. In process, we give a stable rank estimate for maximal full algebras of operator fields over metric spaces.     

Operator space structure on Feichtinger's Segal algebra

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger's remarkable Segal algebra S₀(G). In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow S₀(G) with an operator space structure. With this structure S₀(G) is simultaneously an operator Segal algebra of the Fourier algebra A(G), and of the group algebra L¹(G). We show that this operator space structure is consistent with the major functorial properties: (i) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map is completely surjective, if H is a closed subgroup; and (iii) is completely surjective, where N is a normal subgroup and . We also show that S₀(G) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra.     

Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras

A nonlinear map φ between operator algebras is said to be a numerical radius isometry if w(φ(T−S))=w(T−S) for all T, S in its domain algebra, where w(T) stands for the numerical radius of T. Let and be two atomic nests on complex Hilbert spaces H and K, respectively. Denote the nest algebra associated with and the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from onto and a thorough classification of numerical radius isometries from onto .     

K-group and similarity classification of operators

Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be a Cowen-Douglas operator if there exist Ω, a connected open subset of complex plane C, and n, a positive integer, such that

(a)

(b)

  for z in Ω;

(c)

; and

(d)

for z in Ω.

In the paper, we give a similarity classification of Cowen-Douglas operators by using the ordered K-group of the commutant algebra as an invariant, and characterize the maximal ideals of the commutant algebras of Cowen-Douglas operators. The theorem greatly generalizes the main result in (Canada J. Math. 156(4) (2004) 742) by simply removing the restriction of strong irreducibility of the operators. The research is also partially inspired by the recent classification theory of simple AH algebras of Elliott-Gong in (Documenta Math. 7 (2002) 255; On the classification of simple inductive limit C^*-algebras, II: The isomorphism theorem, preprint.) (also see (J. Funct. Anal. (1998) 1; Ann. Math. 144 (1996) 497; Amer. J. Math. (1996) 187)).     

Additivity of Jordan maps on standard Jordan operator algebras

Let A be a standard Jordan operator algebra on a Hilbert space of dimension >1 and B be an arbitrary Jordan algebra. In this note, we prove that if a bijection ?:A→B satisfies

     

Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization

We consider the Tikhonov-like dynamics where A is a maximal monotone operator on a Hilbert space and the parameter function ε(t) tends to 0 as t→∞ with . When A is the subdifferential of a closed proper convex function f, we establish strong convergence of u(t) towards the least-norm minimizer of f. In the general case we prove strong convergence towards the least-norm point in A⁻¹(0) provided that the function ε(t) has bounded variation, and provide a counterexample when this property fails.     

γ-Bounded representations of amenable groups

Let G be an amenable group, let X be a Banach space and let π:G→B(X) be a bounded representation. We show that if the set is γ-bounded then π extends to a bounded homomorphism w:C^∗(G)→B(X) on the group C^∗-algebra of G. Moreover w is necessarily γ-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G=Z, R or T, and/or when X has property (α).     

Generalized Haar integral

The aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) such that for each f∈A(S) and for any f∈C(Ω,R) and s∈S, , where .     

Fredholm differential operators with unbounded coefficients

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.     

Spin geometry on quantum groups via covariant differential calculi

Let be a cosemisimple Hopf ∗-algebra with antipode S and let Γ be a left-covariant first-order differential ∗-calculus over such that Γ is self-dual (see Section 2) and invariant under the Hopf algebra automorphism S². A quantum Clifford algebra Cl(Γ,σ,g) is introduced which acts on Woronowicz’ external algebra Γ^∧. A minimal left ideal of Cl(Γ,σ,g) which is an -bimodule is called a spinor module. Metrics on spinor modules are investigated. The usual notion of a linear left connection on Γ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac operator and connection Laplacian are defined. For the quantum group SL_q(2) and its bicovariant 4D_±-calculi these concepts are studied in detail. A generalization of Bochner's theorem is given. All invariant differential operators over a given spinor module are determined. The eigenvalues of the Dirac operator are computed.     

Twisted actions and the obstruction to extending unitary representations of subgroups

Suppose that G is a locally compact group and π is a (not necessarily irreducible) unitary representation of a closed normal subgroup N of G on a Hilbert space . We extend results of Clifford and Mackey to determine when π extends to a unitary representation of G on the same space in terms of a cohomological obstruction.     

KKM-Fan’s lemma for solving nonlinear vector evolution equations with nonmonotonic perturbations

In this paper we are interested in the existence of solutions of the following initial value problem: on (0,T) with u(0)=u₀ where A:V→V^′ is a monotone operator, G:V→V^′ is a nonlinear nonmonotone operator and f:(0,T)→V^′ is a measurable function, by means of a recent generalization of the famous KKM-Fan’s lemma.