Equivariant maps and bimodule projections

We construct a counterexample to Solel's [B. Solel, Contractive projections onto bimodules of von Neumann algebras, J. London Math. Soc. 45 (2) (1992) 169–179] conjecture that the range of any contractive, idempotent, MASA bimodule map on is necessarily a ternary subalgebra. Our construction reduces this problem to an analogous problem about the ranges of idempotent maps that are equivariant with respect to a group action. Such maps are important to understand Hamana's theory [M. Hamana, Injective envelopes of C^*-dynamical systems, Tohoku Math. J. 37 (1985) 463–487] of G-injective operator spaces and G-injective envelopes.     

On the diameter of generalized Kneser graphs

Let r, k be positive integers, s(<r), a nonnegative integer, and n=2r-s+k. The set of r-subsets of [n]={1,2,…,n} is denoted by [n]^r. The generalized Kneser graph K(n,r,s) is the graph whose vertex-set is [n]^r where two r-subsets A and B are joined by an edge if |A∩B|?s. This note determines the diameter of generalized Kneser graphs. More precisely, the diameter of K(n,r,s) is equal to , which generalizes a result of Valencia-Pabon and Vera [On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].     

On Polynomial Approximation of Square Integrable Functions on a Subarc of the Unit Circle

Let E be an arc on the unit circle and let L²(E) be the space of all square integrable functions on E. Using the Banach–Steinhaus Theorem and the weak* compactness of the unit ball in the Hardy space, we study the L² approximation of functions in L²(E) by polynomials. In particular, we will investigate the size of the L² norms of the approximating polynomials in the complementary arc E of E. The key theme of this work is to highlight the fact that the benefit of achieving good approximation for a function over the arc E by polynomials is more than offset by the large norms of such approximating polynomials on the complementary arc E.     

On the diameter of a domination dot-critical graph

A graph G is domination dot-critical, or just dot-critical, if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. In this paper, we study an open question concerning of the diameter of a domination dot-critical graph G.     

John's Theorem for an Arbitrary Pair of Convex Bodies

We provide a generalization of John's representation of the identity for the maximal volume position of L inside K, where K and L are arbitrary smooth convex bodies in ⁿ . From this representation we obtain Banach–Mazur distance and volume ratio estimates.     

Cartan–Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems

In this paper, we prove Slodkowski version of the infinite-dimensional spectral mapping theorem and Cartan–Slodkowski version of the finite-dimensional spectral mapping theorem for nilpotent operator Lie subalgebras with respect to the various noncommutative functional calculi.     

A note on isometries of Lipschitz spaces

The main purpose of this article is to generalize a recent result about isometries of Lipschitz spaces. Botelho, Fleming and Jamison [2] described surjective linear isometries between vector-valued Lipschitz spaces under certain conditions. In this article, we extend the main result of [2] by removing the quasi-sub-reflexivity condition from Banach spaces.     

On the diameter of Kronecker graphs

We prove that a.a.s. as soon as a Kronecker graph becomes connected it has a finite diameter.     

Integrability of Analytic Almost Complex Structures on Banach Manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C^*-algebras.Mathematics Subject Classifications (2000): primary 32Q60; secondary 53C15, 58B12.     

On the Asymmetry Constant of a Body with Few Vertices

In this note, we show that a nondegenerated polytope in IR ⁿ with n + k, 1 k < n, vertices is far from any symmetric body. We provide the asymptotically sharp estimates for the asymmetry constant of such polytopes.     

On the connectivity and superconnected graphs with small diameter

In this paper, first we prove that any graph G is 2-connected if diam(G)≤g−1 for even girth g, and for odd girth g and maximum degree Δ≤2δ−1 where δ is the minimum degree. Moreover, we prove that any graph G of diameter diam(G)≤g−2 satisfies that (i) G is 5-connected for even girth g and Δ≤2δ−5, and (ii) G is super-κ for odd girth g and Δ≤3δ/2−1.     

Minimizing the least eigenvalue of unicyclic graphs with fixed diameter

Let U(n,d) be the set of unicyclic graphs on n vertices with diameter d. In this article, we determine the unique graph with minimal least eigenvalue among all graphs in U(n,d). It is found that the extremal graph is different from that for the corresponding problem on maximal eigenvalue as done by Liu et al. [H.Q. Liu, M. Lu, F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007) 449-457].     

A lattice-valued Banach-Stone theorem

Let X and Y be compact Hausdorff spaces, and let E be a Banach lattice. In this short note, we show that if there exists a Riesz isomorphismΦ:C(X, E) →C(Y, R) such that Φ(f) has no zeros if f has none, then X is homeomorphic to Y and E is Riesz isomorphic to R. This revised version was published online in June 2006 with corrections to the Cover Date.     

Total domination in planar graphs of diameter two

MacGillivary and Seyffarth [G. MacGillivray, K. Seyffarth, Domination numbers of planar graphs, J. Graph Theory 22 (1996) 213–229] proved that planar graphs of diameter two have domination number at most three. Goddard and Henning [W. Goddard, M.A. Henning, Domination in planar graphs with small diameter, J. Graph Theory 40 (2002) 1–25] showed that there is a unique planar graph of diameter two with domination number three. It follows that the total domination number of a planar graph of diameter two is at most three. In this paper, we consider the problem of characterizing planar graphs with diameter two and total domination number three. We say that a graph satisfies the domination-cycle property if there is some minimum dominating set of the graph not contained in any induced 5-cycle. We characterize the planar graphs with diameter two and total domination number three that satisfy the domination-cycle property and show that there are exactly thirty-four such planar graphs.     

On diameter and inverse degree of a graph

The inverse degree r(G) of a finite graph G=(V,E) is defined as , where is the degree of vertex v. We establish inequalities concerning the sum of the diameter and the inverse degree of a graph which for the most part are tight. We also find upper bounds on the diameter of a graph in terms of its inverse degree for several important classes of graphs. For these classes, our results improve bounds by Erd?s et al. (1988) [5], and by Dankelmann et al. (2008) [4].     

Lower bounds of the Laplacian spectrum of graphs based on diameter

Let G be a connected graph of order n. The diameter of G is the maximum distance between any two vertices of G. In the paper, we will give some lower bounds for the algebraic connectivity and the Laplacian spectral radius of G in terms of the diameter of G.     

Comparison of Some Distances Between Sums of ℓ n p Spaces

In this paper, we study the classical, modified, and weak Banach–Mazur distances between sums of _p ⁿ spaces. We explicitly calculate the classical and weak Banach–Mazur distances between sums of _p ⁿ spaces and establish bounds for the ratios of these distances.