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Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p × p unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v , satisfy u p = v p = 1 p (the p × p identity matrix) and the commutation relation u v = ζ v u , where ζ is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d × d unitary matrices such that u p = v p = 1 d and u v = ζ v u , then there exists a unital completely positive linear map ϕ : M p ( C ) → M d ( C ) such that ϕ ( u ) = u and ϕ ( v ) = v . We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails. There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63] . This standard construction is generalised herein to the case p ≥ 3 , producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

This paper is a study of linear spaces of matrices and linear maps on matrix algebras that arise from spin systems, or spin unitaries, which are finite sets S of selfadjoint unitary matrices such that any two unitaries in S anticommute. We are especially interested in linear isomorphisms between these linear spaces of matrices such that the matricial order within these spaces is preserved; such isomorphisms are called complete order isomorphisms, which might be viewed as weaker notion of unitary similarity. The main result of this paper shows that all m-tuples of anticommuting selfadjoint unitary matrices are equivalent in this sense, meaning that there exists a unital complete order isomorphism between the unital linear subspaces that these tuples generate. We also show that the C ⁎ -envelope of any operator system generated by a spin system of cardinality 2k or 2 k + 1 is the simple matrix algebra M 2 k ( C ) . As an application of the main result, we show that the free spectrahedra determined by spin unitaries depend only upon the number of the unitaries, not upon the particular choice of unitaries, and we give a new, direct proof of the fact [13] that the spin ball B m spin and max ball B m max coincide as matrix convex sets in the cases m = 1 , 2 . We also derive analogous results for countable spin systems and their C ⁎ -envelopes.

A recent paper of A.~Connes and W.D.~van Suijlekom identifies the operator system of $n\times n$ Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than $n$. The present paper examines this identification in somewhat more detail by showing explicitly that the Connes--van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Consequences of this complete order isomorphism are also examined, yielding two special results of note: (i) that every positive linear map of the $n\times n$ complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the $n\times n$ Toeplitz matrices into the algebra of all $n\times n$ complex matrices is a unitary similarity transformation. This latter result gives a new proof of a theorem established earlier. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of $n\times n$ complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix generates an extremal ray in the cone of all continuous $n\times n$ Toeplitz-matrix valued functions $f$ on the unit circle $S^1$ whose Fourier coffecients $\hat f(k)$ vanish for $|k|\geq n$ ., *** Notational issues corrected in this version

We study the issue of issue of purity (as a completely positive linear map) for identity maps on operators systems and for their completely isometric embeddings into their C$^*$-envelopes and injective envelopes. Our most general result states that the canonical embedding of an operator system $\mathcal R$ into its injective envelope $\mathcal I(\mathcal R)$ is pure if and only if the C$^*$-envelope of $\mathcal R$ is a prime C$^*$-algebra. To prove this, we also show that the identity map on any AW$^*$-factor is a pure completely positive linear map. For embeddings of operator systems into their C$^*$-envelopes, the issue of purity is seemingly harder to describe in full generality, and so we focus here on operator systems arising from the generators of discrete groups. The question of purity of the identity is quite subtle for operator system that are not C$^*$-algebras, and we have results only for two universal operator systems arising from discrete groups. Lastly, a previously unrecorded feature of pure completely positive linear maps is presented: every pure completely positive linear map on an operator system $\mathcal R$ into an injective von Neumann algebra $\mathcal M$ has a pure completely positive extension to any operator system $\mathcal T$ that contains $\mathcal R$ as an operator subsystem, thereby generalising a result of Arveson for the injective type I factor $\mathcal B(\mathcal H)$.

ispartof: Linear Algebra and Its Applications vol:542 pages:1-3 status: published

A classic theorem of T. Ando characterises operators that have numerical radius at most one as operators that admit a certain positive 2x2 operator matrix completion. In this paper we consider variants of Ando's theorem, in which the operators (and matrix completions) are constrained to a given C*-algebra. By considering nxn matrix completions, an extension of Ando's theorem to a multivariable setting is made. We show that the C*-algebras in which these extended formulations of Ando's theorem hold true are precisely the C*-algebras with the weak expectation property (WEP). We also show that a C*-subalgebra A of B(H) has WEP if and only if whenever a certain 3x3 (operator) matrix completion problem can be solved in matrices over B(H), it can also be solved in matrices over A. This last result gives a characterisation of WEP that is spatial and yet is independent of the particular representation of the C*-algebra. This leads to a new characterisation of injective von Neumann algebras. We also give a new equivalent formulation of the Connes Embedding Problem as a problem concerning 3x3 matrix completions., To appear in Journal of Operator Theory

The C*-envelope of the 3-dimensional operator system generated by an irreducible periodic weighted unilateral shift operator is determined.

Fil: Massey, Pedro Gustavo. Universidad Nacional de la Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica; Argentina

The theory of bounded linear operators acting on Hilbert spaces has a special place in functional analysis. In many regards, it is a very specialised part of the subject; yet, it is impressively rich in both theory and application. While the results already established for operators acting on Banach spaces apply automatically to Hilbert space operators, there is at least one aspect in which there is a slight but important departure from Banach space operator theory, and it is the first issue addressed in the present chapter.

A Banach algebra is an associative algebra A with a norm ∥ ⋅ ∥ such that A is a Banach space in this norm, and the norm satisfies ∥ xy ∥ ≤ ∥ x ∥ ∥ y ∥ on all products xy of elements x, y ∈ A. Two examples of Banach algebras encountered to this point are: (i) C(X), the algebra of continuous functions on a compact Hausdorff space X, and (ii) B(V ), the algebra of all bounded linear operators acting on V.

Generic features of topological spaces and their continuous maps were considered in the previous chapter. This chapter investigates certain qualitative features of topological spaces: compactness (how small is a space?), normality (how separated can disjoint closed sets in a space be?), second countability (what is the smallest cardinality of the basis for the topology?), and connectedness (how disperse or disjoint is a space?).

Collections of Hilbert space operators lead to a rich palette of algebraic structures. Of principal interest in this chapter are certain associative algebras of operators, called ∗-algebras, that are closed under the involution T ↦ T∗. However, one could also quite readily consider other algebraic structures, such as semigroups of operators, Lie algebras of operators, or vector spaces of operators. Our focus on ∗-algebras of Hilbert space operators (and their abstractions known as C∗-algebras) stems from the fact that such algebras are widely employed and studied, and exhibit special features that are not present in more generic algebraic structures. The monograph of Paulsen [42] has a good treatment of the theory of general operator algebras and discusses a wide variety of applications to operator theory.

If Banach spaces are viewed as the metric analogue of the notion of vector space, then the concept of an operator is correspondingly viewed as the continuous analogue of the notion of a linear transformation of a vector space.

This paper is an exposition of W.B. Arveson’s complete invariant for the unitary similarity of complex, irreducible matrices.

We present necessary and sufficient conditions for an n\times n complex matrix B to be unitarily similar to a fixed unicellular (i.e., indecomposable by similarity) n\times n complex matrix A, Comment: 16 pages

A precise description of the injective envelope of a spatial continuous trace C*-algebra A over a Stonean space Delta is given. The description is based on the notion of a weakly continuous Hilbert bundle, which we show to be a Kaplansky--Hilbert module over the abelian AW*-algebra C(Delta). We then use the description of the injective envelope of A to study the first- and second-order local multiplier algebras of A. In particular, we show that the second-order local multiplier algebra of A is precisely the injective envelope of A., Comment: 27 pages; many small corrections

The Witt Extension Theorem states that the unitary group of a finite-dimensional vector space V equipped with a nondegenerate hermitian form acts transitively on the pseudosphere induced by the form. We provide a new, constructive proof of this result for finite-dimensional vector spaces V over R, C, or H. This constructive proof is then used to prove a similar result for the unitary group of a finitely generated free right module over an abelian AW∗-algebra. The topology of these unitary groups is examined and as an application we determine the homotopy groups π1 and π2 of the induced real, complex, and quaternionic pseudospheres.

A Bochner-integral formulation of Jensen's inequality is presented for Hermitian matrix-valued functions and measures.

We study the structure of isometries defined on the algebra A of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative norm preserving map A → M n ( C ) must be of the form either A ↦ U A U ⁎ or A ↦ U A ‾ U ⁎ , where A ‾ is the complex conjugation and U is a unitary matrix. In our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry A → M n ( C ) is of the form A ↦ U A V where U and V are two unitary matrices. This implies, in particular, that every such an isometry is a complete isometry and that a unital linear isometry A → M n ( C ) is necessarily an algebra homomorphism.

An operator system is a complex matricially ordered vector space that is completely order isomorphic to a unital selfadjoint subspace of a unital C*-algebra. A matrix state on an operator system V is a unital completely positive linear map of V into a full matrix algebra. Pure matrix states are studied, and a new and somewhat simplified proof of a Krein–Milman-type theorem of Webster and Winkler is given. If V is 3-dimensional, then the matrix state space of V is matrix-affinely homeomorphic to the matricial range of some Hilbert space operator. With the aid of this representation, pure matrix states on 3-dimensional operator systems are examined––and in some cases completely determined.

An exposition of the spectral theory of normal matrices with quaternion entries is presented.

If a and b are trace-class operators, and if u is a partial isometry, then , where ∥⋅∥1 denotes the norm in the trace class. The present paper characterises the cases of equality in this Young inequality, and the characterisation is examined in the context of both the operator and the Hilbert–Schmidt forms of Young's inequality., Facultad de Ciencias Exactas

Josza's definition of fidelity for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C$^*$-algebras $A$ that possess a faithful trace functional $\tau$. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements $\rho\in A$ for which $\tau(\rho)=1$. The second of our two settings is more operator theoretic: by fixing a faithful normal semifinite trace $\tau$ on a semifinite von Neumann algebra $M$, we define and consider the fidelity of pairs of positive operators in $M$ of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of $A$ or of the predual $M_*$. Our results also yield a new proof of a theorem of Moln\'ar on the structure of quantum channels on the trace-class operators that preserve fidelity., Comment: To appear in Journal of Mathematical Physics

If A is a unital C*-algebra and if H is a complex Hilbert space, then the set SH(A) of all unital completely positive linear maps from A to the algebra B(H) of continuous linear operators on H is an operator-valued, or generalised, state space of A. The usual state space of A occurs with the one-dimensional Hilbert space C. The structure of the extreme points of generalised state spaces was determined several years ago by Arveson [Acta Math. 123(1969), 141-224]. Recently, Farenick and Morenz [Trans. Amer. Math. Soc. 349(1997), 1725-1748] studied generalised state spaces from the perspective of noncommutative convexity, and they obtained a number of results on the structure of C*-extreme points. This work is continued in the present paper, and the main result is a precise description of the structure of the C*extreme points of the generalised state spaces of A for all finite-dimensional Hilbert spaces H.

This paper characterises those hyponormal Toeplitz operators on the Hardy space of the unit circle among all Toeplitz operators that have polynomial symbols with circulant-type sets of coefficients.

In this paper we study the space S H ( A ) S_{H}(A) of unital completely positive linear maps from a C ∗ C^{*} -algebra A A to the algebra B ( H ) B(H) of continuous linear operators on a complex Hilbert space H H . The state space of A A , in this notation, is S C ( A ) S_{\mathbb {C}}(A) . The main focus of our study concerns noncommutative convexity. Specifically, we examine the C ∗ C^{*} -extreme points of the C ∗ C^{*} -convex space S H ( A ) S_{H}(A) . General properties of C ∗ C^{*} -extreme points are discussed and a complete description of the set of C ∗ C^{*} -extreme points is given in each of the following cases: (i) the cases S C 2 ( A ) S_{{\mathbb {C}}^{2}}(A) , where A A is arbitrary ; (ii) the cases S C r ( A ) S_{{\mathbb {C}}^{r}}(A) , where A A is commutative; (iii) the cases S C r ( M n ) S_{{\mathbb {C}}^{r}}(M_{n}) , where M n M_{n} is the C ∗ C^{*} -algebra of n × n n\times n complex matrices. An analogue of the Krein-Milman theorem will also be established.

We determine the boundary representations and the C*-envelope of operator systems of the form span{I,T,T*}, where T is a Jordan operator., Comment: 17 pages

It is well known from the work of Brown and Halmos [J. Reine Angew. Math., 213 (1963/1964), pp. 89--102] that an infinite Toeplitz matrix is normal if and only if it is a rotation and translation of a Hermitian Toeplitz matrix. In the present article we prove that all finite normal Toeplitz matrices are either generalised circulants or are obtained from Hermitian Toeplitz matrices by rotation and translation.

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible positive linear maps on von Neumann algebras are explicitly constructed, and a criterion for the irreducibility of decomposable positive maps on full matrix algebras is given.

This paper concerns algebraic and spectral properties of Toeplitz operators T φ T_{\varphi } , on the Hardy space H 2 ( T ) H^{2}({\mathbb {T}}) , under certain assumptions concerning the symbols φ ∈ L ∞ ( T ) \varphi \in L^{\infty }({\mathbb {T}}) . Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at T φ T_{\varphi } , for each quasicontinuous φ \varphi . Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.

A quantum probability measure is a function on a sigma-algebra of subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a measure, but whose values are positive operators acting on a complex Hilbert space, and a quantum random variable is a measurable operator valued function. Although quantum probability measures and random variables are used extensively in quantum mechanics, some of the fundamental probabilistic features of these structures remain to be determined. In this paper we take a step toward a better mathematical understanding of quantum random variables and quantum probability measures by introducing a quantum analogue for the expected value of a quantum random variable relative to a quantum probability measure. In so doing we are led to theorems for a change of quantum measure and a change of quantum variables. We also introduce a quantum conditional expectation which results in quantum versions of some standard identities for Radon-Nikodym derivatives. This allows us to formulate and prove a quantum analogue of Bayes' rule., v2: 16 pages, 0 figures; substantially revised and updated; shortened significantly with streamlined proofs and removal of several explicit calculations

An operator system modulo the kernel of a completely positive linear map of the operator system gives rise to an operator system quotient. In this paper, operator system quotients and quotient maps of certain matrix algebras are considered. Some applications to operator algebra theory are given, including a new proof of Kirchberg's theorem on the tensor product of B(H) with the group C*-algebra of a countable free group. We also show that an affirmative solution to the Connes Embedding Problem is implied by various matrix-theoretic problems, and we give a new characterisation of unital C*-algebras that have the weak expectation property., To appear in Mathematica Scandinavica

The space of positive operator-valued measures on the Borel sets of a compact (or even locally compact) Hausdorff space with values in the algebra of linear operators acting on a d-dimensional Hilbert space is studied from the perspectives of classical and non-classical convexity through a transform $\Gamma$ that associates any positive operator-valued measure with a certain completely positive linear map of the homogeneous C*-algebra $C(X)\otimes B(H)$ into $B(H)$. This association is achieved by using an operator-valued integral in which non-classical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse $\Omega$ for $\Gamma$ yields an integral representation, along the lines of the classical Riesz Representation Theorem for certain linear functionals on $C(X)$, of certain (but not all) unital completely positive linear maps $\phi:C(X)\otimes B(H) \rightarrow B(H)$. The extremal and C*-extremal points of the space of POVMS are determined., Comment: to appear in Journal of Mathematical Physics

In the C ∗ {C^{\ast }} -algebra M n {M_n} of complex n × n n \times n matrices, we consider the notion of noncommutative convexity called C ∗ {C^{\ast }} -convexity and the corresponding notion of a C ∗ {C^{\ast }} -extreme point. We prove that each irreducible element of M n {M_n} is a C ∗ {C^{\ast }} -extreme point of the C ∗ {C^{\ast }} -convex set it generates, and we classify the C ∗ {C^{\ast }} -extreme points of any C ∗ {C^{\ast }} -convex set generated by a compact set of normal matrices.

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A and B be upper triangular n-by-n matrices that (i) are not similar to direct sums of matrices of smaller sizes, or (ii) are in general position and have the same main diagonal. We prove that A and B are unitarily similar if and only if ||h(A_k)||=||h(B_k)|| for all complex polynomials h(x) and k=1, 2, . . , n, where A_k and B_k are the principal k-by-k submatrices of A and B, and ||M|| is the Frobenius norm of M., 16 pages

The local multiplier algebra Mloc(A) of a C*-algebra A has the property that Mloc (A) ⊆ Mloc(Mloc(A)). In this paper we show that there is a separable liminal C*-algebra A such that the inclusion is proper. Fil: Argerami, Martin. University of Regina; Canadá Fil: Farenick, Douglas. University of Regina; Canadá Fil: Massey, Pedro Gustavo. University of Regina; Canadá. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina

A quantum probability measure--or quantum measurement--is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. The notion of a clean quantum measure was introduced by Buscemi et al (2005) for finite-dimensional Hilbert space, and was studied subsequently by Kahn (2007) and Pellonp\"a\"a (2011). The present paper provides new descriptions of clean quantum probability measures in the case of finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of `approximately clean quantum probability measures' and characterise this property for measures whose range determines a finite-dimensional operator system., Comment: 14 pages; fixed/added some details not found in published version

The matrix-valued triangle inequalities of R.C. Thompson [Pacific J. Math. 66 (1976) 285–290] are extended to sequences of matrices with real, complex, or quaternion entries. These new matrix inequalities, in turn, imply a natural formulation of the triangle inequality which is valid in certain Hilbert modules over real or complex semisimple matrix algebras.