Direct Sum Of Matrix Algebras Research Articles

Структура конечной групповой алгебры одного полупрямого произведения абелевых групп и её приложения

В 1978 году Р. Мак-Элисом построена первая асимметричная кодовая криптосистема, основанная на применении помехоустойчивых кодов Гоппы, при этом эффективные атаки на секретный ключ этой криптосистемы до сих пор не найдены. К настоящему времени известно много криптосистем, основанных на теории помехоустойчивого кодирования. Одним из способов построения таких криптосистем является модификация криптосистемы Мак-Элиса с помощью замены кодов Гоппы на другие классы кодов. Однако, известно что криптографическая стойкость многих таких модификаций уступает стойкости классической криптосистемы Мак-Элиса. В связи с развитием квантовых вычислений кодовые криптосистемы, наряду с криптосистемамми на решётках, рассматриваются как альтернатива теоретико-числовым. Поэтому актуальна задача поиска перспективных классов кодов, применимых в криптографии. Представляется, что для этого можно использовать некоммутативные групповые коды, т.е. левые идеалы в конечных некоммутативных групповых алгебрах.Для исследования некоммутативных групповых кодов полезной является теорема Веддерберна, доказывающая существование изоморфизма групповой алгебры на прямую сумму матричных алгебр. Однако конкретный вид слагаемых и конструкция изоморфизма этой теоремой не определены, и поэтому для каждой группы стоит задача конструктивного описания разложения Веддерберна. Это разложение позволяет легко получить все левые идеалы групповой алгебры, т.е. групповые коды. В работе рассматривается полупрямое произведение $$Q_{m,n} = (\mathbb{Z}_m \times \mathbb{Z}_n) \leftthreetimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$$ абелевых групп и конечная групповая алгебра $$\mathbb{F}_q Q_{m,n}$$ этой группы. Для этой алгебры при условиях $$n \mid q -1$$ и $$\text{НОД}(2mn, q) = 1$$ построено разложение Веддербёрна. В случае поля чётной характеристики, когда эта групповая алгебра не является полупростой, также получена сходная структурная теорема. Описаны все неразложимые центральные идемпотенты этой групповой алгебры. Полученные результаты используются для алгебраического описания всех групповых кодов над $$Q_{m,n}.$$

Codes in a Dihedral Group Algebra

In 1978, Robert McEliece constructed the first asymmetric code-based cryptosystem using noise-immune Goppa codes; no effective key attacks has been described for it yet. By now, quite a lot of code-based cryptosystems are known; however, their cryptographic security is inferior to that of the classical McEliece cryptosystem. In connection with the development of quantum computing, code-based cryptosystems are considered as an alternative to number theoretical ones; therefore, the problem of seeking promising classes of codes to construct new secure code-based cryptosystems is relevant. For this purpose, noncommutative codes can be used, that is, ideals in group algebras $${{\mathbb{F}}_{q}}G$$ over finite noncommutative groups $$G$$. The security of cryptosystems based on codes induced by subgroup codes has been studied earlier. The Artin–Wedderburn theorem, which proves the existence of an isomorphism of a group algebra to the direct sum of matrix algebras, is important for studying noncommutative codes. However, the particular form of terms and the construction of the isomorphism are not specified by this theorem; thus, for each group, there remains the problem of constructing the Wedderburn representation. The complete Wedderburn decomposition for the group algebra $${{\mathbb{F}}_{q}}{{D}_{{2n}}}$$ over the dihedral group $${{D}_{{2n}}}$$ has been obtained by F.E. Brochero Martinez in the case when the cardinality of the field and the order of the group are relatively prime numbers. Using these results, we study codes in the group algebra $${{\mathbb{F}}_{q}}{{D}_{{2n}}}$$ in this paper. The problem on the structure of all codes is solved, and the structure of codes induced by codes over cyclic subgroups of $${{D}_{{2n}}}$$ is described, which is of interest for cryptographic applications.

Geometry of C⁎-algebras, and the bidual of their projective tensor product

Given C⁎-algebras A and B, consider the Banach algebra A⊗γB, where ⊗γ denotes the projective Banach space tensor product. If A and B are commutative, this is the Varopoulos algebra VA,B; we write VA for VA,A. It has been an open problem for almost 40 years to determine precisely when A⊗γB is Arens regular; see, e.g., [33], [48], [49]. We solve this classical question for arbitrary C⁎-algebras. Indeed, we show that A⊗γB is Arens regular if and only if A or B has the Phillips property; note that A has the latter property if and only if it is scattered and has the Dunford–Pettis Property. A further equivalent condition is that A⁎ has the Schur property, or, again equivalently, the enveloping von Neumann algebra A⁎⁎ is finite atomic, i.e., a direct sum of matrix algebras. Hence, Arens regularity of A⊗γB is entirely encoded in the geometry of the C⁎-algebras. In case A and B are von Neumann algebra, we conclude that A⊗γB is Arens regular (if and) only if A or B is finite-dimensional. We also show that this characterization does not generalize to the class of non-selfadjoint dual (even commutative) operator algebras. Specializing to commutative C⁎-algebras A and B, we obtain that VA,B is Arens regular if and only if A or B is scattered. We further describe the centre Z(VA⁎⁎), showing that it is Banach algebra isomorphic to A⁎⁎⊗ehA⁎⁎, where ⊗eh denotes the extended Haagerup tensor product. We deduce that VA is strongly Arens irregular (if and) only if A is finite-dimensional. Hence, VA is neither Arens regular nor strongly Arens irregular, if and only if A is non-scattered; this is the case, e.g., for A=ℓ∞.

Codes in Dihedral Group Algebra

Robert McEliece developed an asymmetric encryption algorithm based on the use of binary Goppa codes in 1978 and no effective key attacks has been described yet. Variants of this cryptosystem are known due to the use of different codes types, but most of them were proven to be less secure. Code cryptosystems are considered an alternate to number-theoretical ones in connection with the development of quantum computing. So, the new classes of error-correcting codes are required for building new resistant code cryptosystems. Non-commutative codes, which simply are ideals of finite non-commutative group algebras, are an option. The Artin–Wedderburn theorem implies that a group algebra is isomorphic to a finite direct sum of matrix algebras, when the order of the group and the field characteristics are relatively prime. This theorem is important to study the structure of a non-commutative code, but it gives no information about summands and the isomorphism. In case of a dihedral group these summands and the isomorphism were found by F. E. Brochero Martinez. The purpose of the paper is to study codes in dihedral group algebras as and when the order of a group and a field characteristics are relatively prime. Using the result of F. E. Brochero Martinez, we consider a structure of all dihedral codes in this case and the codes induced by cyclic subgroup codes.

Reduction to Dimension Two of the Local Spectrum for anAHAlgebra with the Ideal Property

AbstractAC*-algebra Ahas the ideal property if any ideal I of Ais generated as a closed two-sided ideal by the projections inside the ideal. Suppose that the limitC*-algebraAof inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has the ideal property. In this paper we will prove thatAcan be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension-drop interval algebras and matrix algebras over 2-dimensional spaces with torsionH2groups.

Affine cellularity of affine Yokonuma–Hecke algebras

We establish an explicit algebra isomorphism between the affine Yokonuma–Hecke algebra Yˆr,n(q) and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type A. As an application of this result, we show that Yˆr,n(q) is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter q is not a root of the Poincaré polynomial. As another application, we also recover the modular representation theory of Yˆr,n(q) previously obtained in [7].

A Kirchberg-type tensor theorem for operator systems

We construct operator systems $\mathfrak C_I$ that are universal in the sense that all operator systems can be realized as their quotients. They satisfy the operator system lifting property. Without relying on the theorem by Kirchberg, we prove the Kirchberg type tensor theorem $$\mathfrak C_I \otimes_{\min} B(H) = \mathfrak C_I \otimes_{\max} B(H).$$ Combining this with a result of Kavruk, we give a new operator system theoretic proof of Kirchberg's theorem and show that Kirchberg's conjecture is equivalent to its operator system analogue $$\mathfrak C_I \otimes_{\min} \mathfrak C_I =\mathfrak C_I \otimes_{\rm c} \mathfrak C_I.$$ It is natural to ask whether the universal operator systems $\mathfrak C_I$ are projective objects in the category of operator systems. We show that an operator system from which all unital completely positive maps into operator system quotients can be lifted is necessarily one-dimensional. Moreover, a finite dimensional operator system satisfying a perturbed lifting property can be represented as the direct sum of matrix algebras. We give an operator system theoretic approach to the Effros-Haagerup lifting theorem.

Representation theory and an isomorphism theorem for the framisation of the Temperley–Lieb algebra

In this paper, we describe the irreducible representations and give a dimension formula for the framisation of the Temperley–Lieb algebra. We then prove that the framisation of the Temperley–Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of classical Temperley–Lieb algebras. This allows us to construct a basis for it. We also study in a similar way the complex reflection Temperley–Lieb algebra.

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

We develop several applications of the fact that the Yokonuma–Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori–Hecke algebras of type A . This includes a description of the semisimple and modular representation theory of the Yokonuma–Hecke algebras of GL and a complete classification of all the Markov traces for them. Finally, from these Markov traces, we construct 3-variables polynomials which are invariants for framed and classical knots and links, and investigate their properties with the help of the isomorphism. In particular, for classical knots, a consequence of the construction is that the obtained set of invariants is topologically equivalent to the HOMFLYPT polynomial. We thus recover results of Chlouveraki et al. (2015, arXiv:1505.06666) about the Juyumaya–Lambropoulou invariants.

Classification of Inductive Limits of Outer Actions of ℝ on Approximate Circle Algebras

AbstractIn this paper we present a classification, up to equivariant isomorphism, of C*-dynamical systems (A, ℝ, α) arising as inductive limits of directed systems ﹛(An, ℝ, αn), φnm﹜, where each An is a finite direct sum of matrix algebras over the continuous functions on the unit circle, and the αns are outer actions generated by rotation of the spectrum.

Triangular [formula omitted]-bialgebra defined as the direct sum of matrix algebras

Let M*(C) denote the C∗-algebra defined as the direct sum of all matrix algebras {Mn(C):n⩾1}. It is known that M*(C) has a non-cocommutative comultiplication Δφ. From a certain set of transformations of integers, we construct a universal R-matrix R of the C∗-bialgebra (M*(C),Δφ) such that the quasi-cocommutative C∗-bialgebra (M*(C),Δφ,R) is triangular. Furthermore, it is shown that certain linear Diophantine equations are corresponded to the Yang–Baxter equations of R.

The Kadison–Singer problem for the direct sum of matrix algebras

Let M n denote the algebra of complex n × n matrices and write M for the direct sum of the M n . So a typical element of M has the form $$ x = x_1\oplus x_2 \cdots \oplus x_n \oplus \cdots, $$where \({x_n \in M_n}\) and \({\|x\| = \sup_n\|x_n\|}\). We set \({D= \{\{x_n\}\in M: x_n\,{\rm is\,diagonal\,for\,all}\,N\}}\). We conjecture (contra Kadison and Singer in Am J Math 81:383–400, 1959) that every pure state of D extends uniquely to a pure state of M. This is known for the normal pure states of D, and we show that this is true for a (weak*) open, dense subset of all the singular pure states of D. We also show that (assuming the Continuum hypothesis) M has pure states that are not multiplicative on any maximal abelian *-subalgebra of M.

A construction of symmetric linear functions on the restricted quantum group $\bar{U}_{q}(\mathit{sl}_{2})$

In this article we construct all the primitive idempotents o f the restricted quantum group Uq (sl2) and also describe Uq (sl2) as the subalgebra of the direct sum of matrix algebras. By using this result we construct a basis of the spa ce of symmetric linear functions of Uq (sl2) and determine the decomposition of the integral of the dual of Uq (sl2) twisted by the balancing element to the basis of the space of symmetric linear functions.

Separable algebras over infinite fields are 2-generated and finitely presented

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M 2(R), under an additional assumption that both pairs generate M 2(R) as an R-algebra.

KK ‐theory of amalgamated free products of C *‐algebras

AbstractJ. Cuntz has conjectured the existence of two cyclic six terms exact sequences relating the KK ‐groups of the amalgamated free product A 1 ∗︁ B A 2 to the KK ‐groups of A 1, A 2 and B. First we establish automatic existence of strict and absorbing homomorphisms. Then we use this result to verify the conjecture when B is a countable direct sum of matrix algebras and the embeddings of B into A 1 and A 2 are quasiunital. Inspired by the proof we achieve the following nice classification result: A separable C *‐algebra B is a countable direct sum of matrix algebras if and only if the unitary group of the multiplier algebra U M (B) is compact in the strict topology. Finally we prove the conjecture when the amalgamated free product has the property that any asymptotically split extension of A 1 ∗︁ B A 2 is split. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Relative amenability and the non-amenability of B(l1)

AbstractIn this paper we begin with a short, direct proof that the Banach algebra B(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebras is no amenable for 1 ≤ p ≤ ∞, p ≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras of B(X) for some space X with symmetric basis—not necessarily X = l2) and we introduce the notion of ‘relative amenability’ of these algebras.

Approximately Finitely Acting Operator Algebras

Let E be an operator algebra on a Hilbert space with finite-dimensional C*-algebra C*(E). A classification is given of the locally finite algebras A0=[formula](Ak, φk) and the operator algebras A=[formula](Ak, φk) obtained as limits of direct sums of matrix algebras over E with respect to star-extendible homomorphisms. The invariants in the algebraic case consist of an additive semigroup, with scale, which is a right module for the semiring VE=Homu(E⊗K, E⊗K) of unitary equivalence classes of star-extendible homomorphisms. This semigroup is referred to as the dimension module invariant. In the operator algebra case the invariants consist of a metrized additive semigroup with scale and a contractive right module VE-action. Subcategories of algebras determined by restricted classes of embeddings, such as 1-decomposable embeddings between digraph algebras, are also classified in terms of simplified dimension modules.

On the classification of simple inductive limit $C^*$-algebras. I: The reduction theorem

Suppose that A = \lim\limits_{n\to\infty}(A_n = \bigoplus_{i=1}^{t_n} M_{[n,i]}(C(X_{n,i})), \phi_{n,m}) is a simple C^* -algebra, where X_{n,i} are compact metrizable spaces of uniformly bounded dimensions (this restriction can be relaxed to a condition of very slow dimension growth). It is proved in this article that A can be written as an inductive limit of direct sums of matrix algebras over certain special 3-dimensional spaces. As a consequence it is shown that this class of inductive limit C^* -algebras is classified by the Elliott invariant – consisting of the ordered K-group and the tracial state space – in a subsequent paper joint with G. Elliott and L. Li (Part II of this series). (Note that the C^* -algebras in this class do not enjoy the real rank zero property.)

A Continuous Field of Projectionless C*-Algebras

AbstractWe use some results about stable relations to show that some of the simple, stable, projectionless crossed products of O2 by considered by Kishimoto and Kumjian are inductive limits of type C*-algebras. The type I C*-algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional C*-algebras.

Classification of direct limits of generalized Toeplitz algebras

Let A be an algebra in the class. The invariant consists of the abelian semigroup V (A), the Murry-von Neumann equivalence classes of projections in matrices of A, an abelian semigroup k(A)+, some equivalence classes of homotopy classes of hyponormal partial isometries in matrices of A and a homomorphism d from k(A)+ into V (A). The main result of this paper states that the above invariant, together with the class of the identity, is complete for the class of C∗-algebras that we consider (cf. §5). For the subclass of the above class which consists of direct limits of finite direct sums of matrix algebras over only non-trivial extensions of the above type, the invariants are even simplier—just V (A). We also show that the algebras in the class exhaust all possible invariants (cf. 3.7). Our paper can be viewed as part of the program of classifying “amenable” C∗-algebras initiated by George A. Elliott. The classical model for the program is the classification of AF -algebras by their dimension groups [Ell1]. Elliott proved in [Ell2] that the class of real rank zero AT-algebras, direct limits of finite direct sums of matrix algebras over C(S) of real rank zero, could be classified by the graded group K0⊕K1 with its natural order. Since then a number of classification results appeared ([BEEK], [D1], [D2], [DL1], [DL2], [Ell3], [Ell4], [Ell5], [EE], [EG1], [EG2], [EGL], [EGLP],