Meet-irreducible ideals and representations of limit algebras

In this paper we give criteria for an ideal of a TAF algebra to be meet-irreducible. We show that is meet-irreducible if and only if the C∗-envelope of is primitive. In that case, admits a faithful nest representation which extends to a ∗-representation of the C∗-envelope for . We also characterize the meet-irreducible ideals as the kernels of bounded nest representations; this settles the question of whether the n-primitive and meet-irreducible ideals coincide.     

Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let γ be the Gauss measure on and the Ornstein-Uhlenbeck operator. For every p in [1,∞)?{2}, set , and consider the sector . The main results of this paper are the following. If p is in (1,∞)?{2}, and , i.e., if M is an Lp(γ)uniform spectral multiplier of in our terminology, and M is continuous on , then M extends to a bounded holomorphic function on the sector . Furthermore, if p=1 a spectral multiplier M, continuous on , satisfies the condition if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i·) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on belonging to a wide class, which contains . From these results we deduce that operators in this class do not admit an H^∞ functional calculus in sectors smaller than .     

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 .     

Norm estimates of almost Mathieu operators

We estimate the norm of the almost Mathieu operator , regarded as an element in the rotation C^*-algebra . In the process, we prove for every λ∈R and the inequality

     

The Rokhlin property and the tracial topological rank

Let A be a unital separable simple C∗-algebra  with TR(A)?1 and α be an automorphism. We show that if α satisfies the tracially cyclic Rokhlin property then . We also show that whenever A has a unique tracial state and α^m is uniformly outer for each m(≠0) and α^r is approximately inner for some r>0, α satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear C∗-algebras, we use the above result to prove a conjecture of Kishimoto: if A is a unital simple -algebra of real rank zero and α∈Aut(A) which is approximately inner and if α satisfies some Rokhlin property, then the crossed product is again an -algebra of real rank zero. As a by-product, we find that one can construct a large class of simple C∗-algebras with tracial rank one (and zero) from crossed products.     

Singular-value-like decomposition for complex matrix triples

The classical singular value decomposition for a matrix A∈C^m×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA^∗ and A^∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices and . More generally, we consider the matrix triple , where are invertible and either complex symmetric or complex skew-symmetric, and we provide a canonical form under transformations of the form , where X,Y are nonsingular.     

The structure of a tridiagonal pair

Let K denote a field and let V denote a vector space over K with finite positive dimension.We consider a pair of K-linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W,A^∗W⊆W,W≠0,W≠V.We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide.In this paper we show that the following (i)-(iv) hold provided that K is algebraically closed: (i) Each of has dimension 1.(ii) There exists a nondegenerate symmetric bilinear form 〈,〉 on V such that 〈Au,v〉=〈u,Av〉 and 〈A^∗u,v〉=〈u,A^∗v〉 for all u,v∈V.(iii) There exists a unique anti-automorphism of End(V) that fixes each of A,A^∗.(iv) The pair A,A^∗ is determined up to isomorphism by the data , where θ_i (resp.) is the eigenvalue of A (resp.A^∗) on V_i (resp.), and is the split sequence of A,A^∗ corresponding to and .     

Positivity properties in noncommutative convolution algebras with applications in pseudo-differential calculus

We study , of all such that for every ?∈C^∞₀, where denotes the twisted convolution. We prove that certain boundedness for are completely determined of the behaviour for a at origin, for example that , and that if a(0)<∞, then a∈L²∩L^∞. We use the results in order to determine wether positive pseudo-differential operators belong to certain Schatten-casses or not.     

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 duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals

For a locally compact group G, let XG be one of the following introverted subspaces of VN(G): , the C^∗-algebra of uniformly continuous functionals on A(G); , the space of weakly almost periodic functionals on A(G); or , the C^∗-algebra generated by the left regular representation on the measure algebra of G. We discuss the extension of homomorphisms of (reduced) Fourier-Stieltjes algebras on G and H to cb-norm preserving, weak^∗-weak^∗-continuous homomorphisms of into , where (XG,XH) is one of the pairs , , or . When G is amenable, these extensions are characterized in terms of piecewise affine maps.     

Solution to a conjecture by Hofmeier-Wittstock

For a locally compact (non-compact) group , consider as a left module over the algebra , endowed with the first Arens product. We answer in the affirmative a question raised by Hofmeier-Wittstock (Math. Ann. 308 (1) (1997) 141) concerning the automatic boundedness of the corresponding module homomorphisms on . In fact, we prove that an even stronger assertion holds true. Furthermore, we show that the theorem, first obtained by Ghahramani-McClure (Canad. Math. Bull. 35 (2) (1992) 180), on the automatic w∗-continuity of the (bounded) -module homomorphisms on , can be sharpened in a similar fashion. To this end, a general factorization theorem for bounded families in is proved from which we shall derive both results.     

Reflection about k-jets and holomorphic extension of CR mappings

Let be a CR mapping between real analytic generic submanifolds M, M₁ of and , respectively. According to Webster's theory (Proc. Amer. Math. Soc. 86 (1982) 236-240) and its further developments, f has holomorphic extension to a full neighborhood of M in when the following requirements are fulfilled: f extends to a wedge W continuous up to M; f is of class C^k; (where denotes the complex tangent bundle); M₁ is “k-nondegenerate.” We deal here with the case where is strictly smaller than but is still real analytic in suitable sense. We show that a suitably refined condition of k-nondegeneracy still entails holomorphic extension of f.     

The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .     

Generalized vector valued almost periodic and ergodic distributions

Schwartz's almost periodic distributions are generalized to the case of Banach space valued distributions , and furthermore for a given arbitrary class A to for φ∈ test functions D(R,C)}. It is shown that this extension process is iteration complete, i.e. . Moreover the T from are characterized in various ways, also tempered distributions with P={X-valued functions of polynomial growth} are shown. Under suitable assumptions , , where for all h>0}, is defined with the corresponding extension of Mh. With an extension of the indefinite integral from to D^′(R,X) a distribution analogue to the Bohl-Bohr-Amerio-Kadets theorem on the almost periodicity of bounded indefinite integrals of almost periodic functions is obtained, also for almost automorphic, Levitan almost periodic and recurrent functions, similar for a result of Levitan concerning ergodic indefinite integrals. For many of the above results a new (Δ)-condition is needed, we show that it holds for most of the A needed in applications. Also an application to the study of asymptotic behavior of distribution solutions of neutral integro-differential-difference systems is given.     

Classification of contractively complemented Hilbertian operator spaces

We construct some separable infinite-dimensional homogeneous Hilbertian operator spaces and , which generalize the row and column spaces R and C (the case m=0). We show that a separable infinite-dimensional Hilbertian JC^∗-triple is completely isometric to one of , , , or the space Φ spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that (respectively ) is completely isometric to the space of creation (respectively annihilation) operators on the m (respectively m+1) anti-symmetric tensors of the Hilbert space. Together with the finite-dimensional case studied in [M. Neal, B. Russo, Representation of contractively complemented Hilbertian operator spaces on the Fock space, Proc. Amer. Math. Soc. 134 (2006) 475-485], this gives a full operator space classification of all rank-one JC^∗-triples in terms of creation and annihilation operator spaces.We use the above structural result for Hilbertian JC^∗-triples to show that all contractive projections on a C^∗-algebra A with infinite-dimensional Hilbertian range are “expansions” (which we define precisely) of normal contractive projections from A** onto a Hilbertian space which is completely isometric to R, C, R∩C, or Φ. This generalizes the well-known result, first proved for B(H) by Robertson in [A.G. Robertson, Injective matricial Hilbert spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991) 183-190], that all Hilbertian operator spaces that are completely contractively complemented in a C^∗-algebra are completely isometric to R or C. We use the above representation on the Fock space to compute various completely bounded Banach-Mazur distances between these spaces, or Φ.