Extension algebras of Cuntz algebra

In this paper, we construct representatives for all equivalence classes of the unital essential extension algebras of Cuntz algebra by the C^*-algebras of compact operators on a separable infinite-dimensional Hilbert space. We also compute their K-groups and semigroups and classify these extension algebras up to isomorphism by their semigroups.     

A crossed product of the canonical anticommutation relations algebra in the Cuntz algebra

We show that the Cuntz algebra can be represented as a crossed product of the canonical anticommutation relations algebra by an endomorphism.     

The cyclic and simplicial cohomology of the Cuntz semigroup algebra

The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra ${?}^1({\mathbf{S}}_m)$. We also determine the simplicial and cyclic cohomology of the tensor algebra of a Banach space, a class which includes the algebra on the free semigroup on m-generators ${?}^1({\mathbf{FS}}_m)$. In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to ${?}^1({\mathbf{S}}_m)$, showing that the cyclic cohomology groups of degree n vanish when n is odd and are one-dimensional when n is even ($n?2$). Using the Connes–Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree n vanish for $n?1$. A similar strategy is used for the tensor algebra of a Banach space.     

Weakly almost periodic functionals on the measure algebra

It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C*-algebra. This implies that the weakly almost periodic functionals on M(G), the measure algebra of a locally compact group G, is a C*-subalgebra of M(G)* = C ₀(G)**. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps. The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces.     

The positivity of linear functionals on Cuntz algebras associated to unit vectors

We study the linear functional on the Cuntz algebra associated to a sequence of unit vectors in that is a generalization of the Cuntz state. We prove that is positive if and only if is a constant sequence.

     

赋范线性空间上有界线性泛函的一般形式

1968年,N.E.Gretsky求得了Banach函数空间上有界线性泛函的一般形式。1977年,G.D.Faulkner对自反的Banach空间解决了这个问题。又本文作者对一类无限多个Banach空间解决了有界线性泛函的表示问题。 本文对一般的赋范线性空间给出了有界线性泛函的表达式和共轭空间的等距同构表现。     

Linear discrete operators on the disk algebra

Let be the disk algebra. In this paper we address the following question: Under what conditions on the points do there exist operators such that

and , , for every ? Here the convergence is understood in the sense of norm in . Our first result shows that if satisfy Carleson condition, then there exists a function such that , . This is a non-trivial generalization of results of Somorjai (1980) and Partington (1997). It also provides a partial converse to a result of Totik (1984). The second result of this paper shows that if are required to be projections, then for any choice of the operators do not converge to the identity operator. This theorem generalizes the famous theorem of Faber and implies that the disk algebra does not have an interpolating basis.

     

Irreducible representations of the Cuntz algebra

In this paper, we establish formulas for the configuration of a special class of irreducible representations of the Cuntz algebra , . These irreducible representations arise as subrepresentations of naturally occurring representations of acting in and arise from consideration of multiresolution wavelet filters.

     

Linear functionals and the duality principle for harmonic functions

Let ${\mathcal H}$ be the class of complex‐valued harmonic functions in the unit disk |z| < 1 and ${\mathcal H}_1$ the set of all functions $f\in {\mathcal H}$ such that f(0) = 0, f_z(0) = 1 and $f_{\overline{z}}(0)=0$. For $V \subset {\mathcal H}_1$, its dual V* is where * denotes the Hadamard product for harmonic functions. The set V is a dual class if V = W* for some $W \subset {\mathcal H}_1.$ In the present paper, the duality principle is extended to ${\mathcal H}_1$ by means of the Hadamard product. Counterparts of the dual classes are introduced and their structural properties studied.     

Outer Automorphisms on Cuntz Algebras

We prove that if an automorphism on the Cuntz algebra O_n satisfies(S₁) = S₁ for one isometry S₁ of generators S₁, ..., S_n suchthat , , l i n and a complex number with modulus one,then it is outer. In particular, any non trivial automorphismon O_n leaving one generator fixed is outer.     

Linear functionals of eigenvalues of random matrices

Let be a random unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of to converge to a Gaussian limit as . By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of . For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices.

     

Linear algebra methods for forbidden configurations

We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given m and a k×l (0,1)-matrix F we define forb(m,F) as the maximum number of columns in a simple m-rowed matrix A for which no k×l submatrix of A is a row and column permutation of F. In set theory notation, F is a forbidden trace. For all k-rowed F (simple or non-simple) Füredi has shown that forb(m,F) is O(m ^k ). We are able to determine for which k-rowed F we have that forb(m,F) is O(m ^k−1) and for which k-rowed F we have that forb(m,F) is Θ(m ^k ).