On some algebraic properties of locally compact and weakly linearly compact topological Abelian groups

Locally compact and weakly linearly compact topological groups are studied. The notion of a weakly linearly compact topological Abelian group is a generalization of the notion of a weakly separable topological Abelian group, introduced by N. Ya. Vilenkin. Some algebraic properties of these groups are studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 41, Topology and Its Applications, 2006.     

An algebraic study of BIB designs: A complete solution for v = 6 and k = 3

The set of all distinct blocks of a BIB design is referred to as the support of the design. The family of BIB designs with v = 6 and k = 3 is studied in depth from the view of possible support sizes. Via a linear algebraic technique it is shown that: (i) a BIB design with v = 6 and k = 3 exists if and only if the support size belongs to {10, 14, 16, 17, 18, 19, 20}; (ii) the minimum block size, b, needed to obtain a BIB design with support size $\text{b}{}^1$ is given by:

     

4.

Invariant subspaces of compact elements inC *-algebras     

M. R. F. Smyth  T. T. West 《Mathematische Zeitschrift》1977,153(2):193-197

     

5.

Classification of extensions of classifiable -algebras     

Sren Eilers  Gunnar Restorff  Efren Ruiz 《Advances in Mathematics》2009,222(6):2153-2172

For a certain class of extensions of C^*-algebras in which B and A belong to classifiable classes of C^*-algebras, we show that the functor which sends to its associated six term exact sequence in K-theory and the positive cones of K₀(B) and K₀(A) is a classification functor. We give two independent applications addressing the classification of a class of C^*-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C^*-algebras.     

6.

On compact classes of algebraic systems     

S. R. Kogalovskii 《Algebra and Logic》1968,7(2):86-94

     

7.

Semicrossed products of simple C*-algebras     

Kenneth R. Davidson  Elias G. Katsoulis 《Mathematische Annalen》2008,342(3):515-525

Let and be C*-dynamical systems and assume that is a separable simple C*-algebra and that α and β are *-automorphisms. Then the semicrossed products and are isometrically isomorphic if and only if the dynamical systems and are outer conjugate. K. R. Davidson was partially supported by an NSERC grant. E. G. Katsoulis was partially supported by a summer grant from ECU     

8.

N=4 Multiplets in N=3 Harmonic Superspace     

Zupnik  B. M. 《Theoretical and Mathematical Physics》2004,140(2):1121-1134

We show that the N=3 harmonic superfield equations of motion are invariant with respect to the fourth supersymmetry. We also use the SU(3) harmonics to analyze a more flexible form of superfield constraints for the Abelian N=4 vector multiplet and its N=3 decomposition. An unusual alternative representation of the N=4 supersymmetry is realized on infinite multiplets of analytic superfields in the N=3 harmonic superspace. An integer-valued parameter playing the role of a discrete coordinate parameterizes U(1) charges of superfields in these multiplets. Each superfield term of the N=3 Yang–Mills action has an infinite-dimensional N=4 generalization. The gauge group of this model contains an infinite number of superfield parameters.     

9.

C^*-algebras of Toeplitz type associated with algebraic number fields     

Joachim Cuntz  Christopher Deninger  Marcelo Laca 《Mathematische Annalen》2013,355(4):1383-1423

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra ${\mathfrak T }[R]$ . It is an extension of the ring C*-algebra ${\mathfrak A }[R]$ studied previously by the first named author in collaboration with X. Li. In contrast to ${\mathfrak A }[R]$ , it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$ -semigroup $R\rtimes R^\times $ on $\ell ^2 (R\rtimes R^\times )$ . The algebra ${\mathfrak T }[R]$ carries a natural one-parameter automorphism group $(\sigma _t)_{t\in {\mathbb R }}$ . We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The “partition functions” are partial Dedekind $\zeta $ -functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\zeta $ -functions, which we then use to show uniqueness of the $\beta $ -KMS state for each inverse temperature $\beta \in (1,2]$ .     

10.

Factorizations of simple algebraic groups     

Martin W. Liebeck  Jan Saxl  Gary M. Seitz 《Transactions of the American Mathematical Society》1996,348(2):799-822

We determine all factorizations of simple algebraic groups as the product of two maximal closed connected subgroups. Additional results are established which drop the maximality assumption, and applications are given to the study of subgroups of classical groups transitive on subspaces of a given dimension.

     

11.

A non-metrizable compact linearly ordered topological space, every subspace of which has a -minimal base     

Wei-Xue Shi 《Proceedings of the American Mathematical Society》1999,127(9):2783-2791

A collection of subsets of a space is minimal if each element of contains a point which is not contained in any other element of . A base of a topological space is -minimal if it can be written as a union of countably many minimal collections. We will construct a compact linearly ordered space satisfying that is not metrizable and every subspace of has a -minimal base for its relative topology. This answers a problem of Bennett and Lutzer in the negative.

     

12.

The algebraic structure of linearly recursive sequences under hadamard product     

Richard G. Larson  Earl J. Taft 《Israel Journal of Mathematics》1990,72(1-2):118-132

We describe the algebraic structure of linearly recursive sequences under the Hadamard (point-wise) product. We characterize the invertible elements and the zero divisors. Our methods use the Hopf-algebraic structure of this algebra and classical results on Hopf algebras. We show that our criterion for invertibility is effective if one knows a linearly recursive relation for a sequence and certain information about finitely-generated subgroups of the multiplicitive group of the field. Supported in part by NSF Grant DMS 870-1085.     

13.

Type factors generated by purely infinite simple C*-algebras associated with free groups     

Wojciech Szymanski  Shuang Zhang 《Proceedings of the American Mathematical Society》2000,128(3):813-818

Let be a free product of at least two but at most countably many cyclic groups. With each such group we associate a family of C*-algebras, denoted and generated by the reduced group C*-algebra and a collection of projections onto the -spaces over certain subsets of . We determine , the weak closure of in , and use this result to show that many of the C*-algebras in question are non-nuclear.

     

14.

Diagonalization of compact operators in Hilbert modules over finiteW*-algebras     

V. M. Manuilov 《Annals of Global Analysis and Geometry》1995,13(3):207-226

It is known that a continuous family of compact self-adjoint operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators on Hilbert modules over a commutativeW*-algebra. The aim of the present paper is to generalize this fact to a finiteW*-algebraA not necessarily commutative. We prove that for a compact operatorK acting on the right HilbertA-moduleH* _A dual toH _A under slight restrictions one can find a set of eigenvectorsx _i H* _A and a non-increasing sequence of eigenvalues _i A such thatK x _i=x _{i i} and the selfdual HilbertA-module generated by these eigenvectors is the wholeH* _A. As an application we consider the Schrödinger operator in a magnetic field with irrational magnetic flow as an operator acting on a Hilbert module over the irrational rotation algebraA and discuss the possibility of its diagonalization.     

15.

Purely infinite corona algebras of simple C*-algebras     

Dan Kucerovsky  P. W. Ng  Francesc Perera 《Mathematische Annalen》2010,346(1):23-40

In this paper, we study the problem of when the corona algebra of a non-unital C*-algebra is purely infinite. A complete answer is obtained for stabilisations of simple and unital algebras that have enough comparison of positive elements. Our result relates the pure infiniteness condition (from its strongest to weakest forms) to the geometry of the tracial simplex of the algebra, and to the behaviour of corona projections, despite the fact that there is no real rank zero condition.     

16.

On the stable rank of simple C*-algebras     

Jesper Villadsen 《Journal of the American Mathematical Society》1999,12(4):1091-1102

It is shown that there exists a simple, finite C*-algebra whose topological (or Bass) stable rank is any given natural number or .

     

17.

Jordan subgroups of simple algebraic groups     

A. V. Borovik 《Algebra and Logic》1989,28(2):97-108

     

18.

Classification of certain infinite simpleC *-algebras,II     

George A. Elliott  Mikael Rørdam 《Commentarii Mathematici Helvetici》1995,70(1):615-638

     

19.

On the maximal regular ideal of a linearly compact ring     

Dinh Van Huynh 《Archiv der Mathematik》1979,33(1):232-234

     

20.

Automatic surjectivity of ring homomorphisms on -algebras and algebraic differences among some group algebras of compact groups     

Lajos Molná  r 《Proceedings of the American Mathematical Society》2000,128(1):125-134

In this paper we present two automatic surjectivity results concerning ring homomorphisms between -classes of an -algebra which, in some sense, improve the main theorem in a recent paper by the author (Proc. Amer. Math. Soc. 124 (1996), 169-175) quite significantly. Furthermore, we apply our results to show that for arbitrary infinite compact groups , no quotient ring of is isomorphic to , a statement we conjecture to be true for every pair of group rings corresponding to different exponents .

     

$\text{b}{}^1$	10	14	16	17	18	19	20
$\text{b}$	10	20	20	30	40	30	20

Classification of extensions of classifiable -algebras

For a certain class of extensions of C^*-algebras in which B and A belong to classifiable classes of C^*-algebras, we show that the functor which sends to its associated six term exact sequence in K-theory and the positive cones of K₀(B) and K₀(A) is a classification functor. We give two independent applications addressing the classification of a class of C^*-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C^*-algebras.     

Semicrossed products of simple C*-algebras

Let and be C*-dynamical systems and assume that is a separable simple C*-algebra and that α and β are *-automorphisms. Then the semicrossed products and are isometrically isomorphic if and only if the dynamical systems and are outer conjugate. K. R. Davidson was partially supported by an NSERC grant. E. G. Katsoulis was partially supported by a summer grant from ECU     

N=4 Multiplets in N=3 Harmonic Superspace

We show that the N=3 harmonic superfield equations of motion are invariant with respect to the fourth supersymmetry. We also use the SU(3) harmonics to analyze a more flexible form of superfield constraints for the Abelian N=4 vector multiplet and its N=3 decomposition. An unusual alternative representation of the N=4 supersymmetry is realized on infinite multiplets of analytic superfields in the N=3 harmonic superspace. An integer-valued parameter playing the role of a discrete coordinate parameterizes U(1) charges of superfields in these multiplets. Each superfield term of the N=3 Yang–Mills action has an infinite-dimensional N=4 generalization. The gauge group of this model contains an infinite number of superfield parameters.     

C^*-algebras of Toeplitz type associated with algebraic number fields

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra ${\mathfrak T }[R]$ . It is an extension of the ring C*-algebra ${\mathfrak A }[R]$ studied previously by the first named author in collaboration with X. Li. In contrast to ${\mathfrak A }[R]$ , it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$ -semigroup $R\rtimes R^\times $ on $\ell ^2 (R\rtimes R^\times )$ . The algebra ${\mathfrak T }[R]$ carries a natural one-parameter automorphism group $(\sigma _t)_{t\in {\mathbb R }}$ . We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The “partition functions” are partial Dedekind $\zeta $ -functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\zeta $ -functions, which we then use to show uniqueness of the $\beta $ -KMS state for each inverse temperature $\beta \in (1,2]$ .     

Factorizations of simple algebraic groups

We determine all factorizations of simple algebraic groups as the product of two maximal closed connected subgroups. Additional results are established which drop the maximality assumption, and applications are given to the study of subgroups of classical groups transitive on subspaces of a given dimension.

     

A non-metrizable compact linearly ordered topological space, every subspace of which has a -minimal base

A collection of subsets of a space is minimal if each element of contains a point which is not contained in any other element of . A base of a topological space is -minimal if it can be written as a union of countably many minimal collections. We will construct a compact linearly ordered space satisfying that is not metrizable and every subspace of has a -minimal base for its relative topology. This answers a problem of Bennett and Lutzer in the negative.

     

The algebraic structure of linearly recursive sequences under hadamard product

We describe the algebraic structure of linearly recursive sequences under the Hadamard (point-wise) product. We characterize the invertible elements and the zero divisors. Our methods use the Hopf-algebraic structure of this algebra and classical results on Hopf algebras. We show that our criterion for invertibility is effective if one knows a linearly recursive relation for a sequence and certain information about finitely-generated subgroups of the multiplicitive group of the field. Supported in part by NSF Grant DMS 870-1085.     

Type factors generated by purely infinite simple C*-algebras associated with free groups

Let be a free product of at least two but at most countably many cyclic groups. With each such group we associate a family of C*-algebras, denoted and generated by the reduced group C*-algebra and a collection of projections onto the -spaces over certain subsets of . We determine , the weak closure of in , and use this result to show that many of the C*-algebras in question are non-nuclear.

     

Diagonalization of compact operators in Hilbert modules over finiteW*-algebras

It is known that a continuous family of compact self-adjoint operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators on Hilbert modules over a commutativeW*-algebra. The aim of the present paper is to generalize this fact to a finiteW*-algebraA not necessarily commutative. We prove that for a compact operatorK acting on the right HilbertA-moduleH* _A dual toH _A under slight restrictions one can find a set of eigenvectorsx _i H* _A and a non-increasing sequence of eigenvalues _i A such thatK x _i=x _{i i} and the selfdual HilbertA-module generated by these eigenvectors is the wholeH* _A. As an application we consider the Schrödinger operator in a magnetic field with irrational magnetic flow as an operator acting on a Hilbert module over the irrational rotation algebraA and discuss the possibility of its diagonalization.     

Purely infinite corona algebras of simple C*-algebras

In this paper, we study the problem of when the corona algebra of a non-unital C*-algebra is purely infinite. A complete answer is obtained for stabilisations of simple and unital algebras that have enough comparison of positive elements. Our result relates the pure infiniteness condition (from its strongest to weakest forms) to the geometry of the tracial simplex of the algebra, and to the behaviour of corona projections, despite the fact that there is no real rank zero condition.     

On the stable rank of simple C*-algebras

It is shown that there exists a simple, finite C*-algebra whose topological (or Bass) stable rank is any given natural number or .

     

Automatic surjectivity of ring homomorphisms on -algebras and algebraic differences among some group algebras of compact groups

In this paper we present two automatic surjectivity results concerning ring homomorphisms between -classes of an -algebra which, in some sense, improve the main theorem in a recent paper by the author (Proc. Amer. Math. Soc. 124 (1996), 169-175) quite significantly. Furthermore, we apply our results to show that for arbitrary infinite compact groups , no quotient ring of is isomorphic to , a statement we conjecture to be true for every pair of group rings corresponding to different exponents .