Topological free entropy dimension in unital C-algebras

The notion of topological free entropy dimension of n-tuple of elements in a unital C^∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint element in a unital C^∗ algebra. We also calculate the values of topological free entropy dimensions of any families of self-adjoint generators of some unital C^∗ algebras, including irrational rotation C^∗ algebra, UHF algebra, and minimal tensor product of two reduced C^∗ algebras of free groups.     

Approximate and pseudo-amenability of various classes of Banach algebras

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of ?¹-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups.     

Growth Behavior of Two Classes of Merit Functions for Symmetric Cone Complementarity Problems

In the solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit Lagrangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the P-property; for the general SCCP, the EP merit functions are coercive only if the underlying mapping has the uniform Jordan P-property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the mapping, for example, the Lipschitz continuity or the assumption as in (45). The authors would like to thank the two anonymous referees for their helpful comments which improved the presentation of this paper greatly. The research of J.-S. Chen was partially supported by National Science Council of Taiwan.     

The clique numbers of regular graphs of matrix algebras are finite

Let F be a field, char(F)≠2, and SGL_n(F), where n is a positive integer. In this paper we show that if for every distinct elements x,yS, x+y is singular, then S is finite. We conjecture that this result is true if one replaces field with a division ring.     

One more simple proof of the Craig–Sakamoto theorem

We give one more elementary proof of the Craig-Sakamoto’s theorem: given such that ; then AB=0.     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

Continuity and generators of dynamical semigroups for infinite Bose systems

For a class of quasifree quantum dynamical semigroups on the algebra of the canonical commutation relations (CCR) we give sufficient conditions for these semigroups to extend to ultraweakly continuous semigroups of normal operators on the von Neumann algebra associated with a representation of the CCR. Then the explicit form of the generators of the extended semigroups is calculated.     

Sufficient conditions for the projective freeness of Banach algebras

Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H^∞(X) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.     

The universal modality, the center of a Heyting algebra, and the Blok–Esakia theorem

We introduce the bimodal logic , which is the extension of Bennett’s bimodal logic by Grzegorczyk’s axiom □(□(p→□p)→p)→p and show that the lattice of normal extensions of the intuitionistic modal logic WS5 is isomorphic to the lattice of normal extensions of , thus generalizing the Blok–Esakia theorem. We also introduce the intuitionistic modal logic WS5.C, which is the extension of WS5 by the axiom (p¬p)→(p→p), and the bimodal logic , which is the extension of Shehtman’s bimodal logic by Grzegorczyk’s axiom, and show that the lattice of normal extensions of WS5.C is isomorphic to the lattice of normal extensions of .     

Weak Quantum Enveloping Algebras of Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras . It is a noncommutative and noncocommutative weak graded Hopf algebra. Using localizing with some Ore set, we obtain a different kind of quantized enveloping algebras of Borcherds superalgebras . It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra of . Moreover, is isomorphic to a direct sum of and an other algebra as algebras. The author is sponsored by ZJNSF No. Y607136.