A Generalization of Strassen’s Theorem on Preordered Semirings

Given a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

CLOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT OF CERTAIN C*-ALGEBRAS

For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.

Chebyshev-Type Integral Inequalities for Continuous Fields of Operators Concerning Khatri–Rao Products and Synchronous Properties

We consider bounded continuous fields of self-adjoint operators which are parametrized by a locally compact Hausdorff space Ω equipped with a finite Radon measure μ . Under certain assumptions on synchronous Khatri–Rao property of the fields of operators, we obtain Chebyshev-type inequalities concerning Khatri–Rao products. We also establish Chebyshev-type inequalities involving Khatri–Rao products and weighted Pythagorean means under certain assumptions of synchronous monotone property of the fields of operators. The Pythagorean means considered here are three classical symmetric means: the geometric mean, the arithmetic mean, and the harmonic mean. Moreover, we derive the Chebyshev–Grüss integral inequality via oscillations when μ is a probability Radon measure. These integral inequalities can be reduced to discrete inequalities by setting Ω to be a finite space equipped with the counting measure. Our results provide analog results for matrices and integrable functions. Furthermore, our results include the results for tensor products of operators, and Khatri–Rao/Kronecker/Hadamard products of matrices, which have been not investigated in the literature.

Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

Integral Inequalities of Chebyshev Type for Continuous Fields of Hermitian Operators Involving Tracy–Singh Products and Weighted Pythagorean Means

In this paper, we establish several integral inequalities of Chebyshev type for bounded continuous fields of Hermitian operators concerning Tracy-Singh products and weighted Pythagorean means. The weighted Pythagorean means considered here are parametrization versions of three symmetric means: the arithmetic mean, the geometric mean, and the harmonic mean. Every continuous field considered here is parametrized by a locally compact Hausdorff space equipped with a finite Radon measure. Tracy-Singh product versions of the Chebyshev-Grüss inequality via oscillations are also obtained. Such integral inequalities reduce to discrete inequalities when the space is a finite space equipped with the counting measure. Moreover, our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.

ON A FUNCTION MODULE WITH APPROXIMATE HYPERPLANE SERIES PROPERTY

We present a sufficient and necessary condition for a function module space $X$ to have the approximate hyperplane series property (AHSP). As a consequence, we have that the space ${\mathcal{C}}_{0}(L,E)$ of bounded and continuous $E$-valued mappings defined on the locally compact Hausdorff space $L$ has AHSP if and only if $E$ has AHSP.

On dimension andweight of a local contact algebra

As proved in [16], there exists a duality ?t between the category HLC of locally compact Hausdorff spaces and continuous maps, and the category DHLC of complete local contact algebras and appropriate morphisms between them. In this paper, we introduce the notions of weight wa and of dimension dima of a local contact algebra, and we prove that if X is a locally compact Hausdorff space then w(X) = wa(?t(X)), and if, in addition, X is normal, then dim(X) = dima(?t(X)).

The Weak Ideal Property and Topological Dimension Zero

Following up on previous work, we prove a number of results for C* -algebras with the weak ideal property or topological dimension zero, and some results for C* -algebras with related properties. Some of the more important results include the following:The weak ideal property implies topological dimension zero.For a separable C* -algebra A, topological dimension zero is equivalent to , to D ⊗ A having the ideal property for some (or any) Kirchberg algebra D, and to A being residually hereditarily in the class of all C* -algebras B such that contains a nonzero projection.Extending the known result for , the classes of C* -algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian 2-groups.If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A ⊗ B has the weak ideal property.If X is a totally disconnected locally compact Hausdorff space and A is a C0(X)-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for topological dimension zero, provided A is separable).Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C* -algebras, including all separable locally AH algebras.The weak ideal property does not imply the ideal property for separable Z-stable C* -algebras.We give other related results, as well as counterexamples to several other statements one might conjecture.

On the decomposition into Discrete, Type II and Type III C*-algebras

We obtained a “decomposition scheme” of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsidó), type II C*-algebras and type III C*-algebras (both defined by Cuntz and Pedersen) form a good framework to “classify” C*-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C*-subalgebras as well as taking “essential extension” and “normal quotient”. Furthermore, there exist the largest discrete finite ideal Ad,1, the largest discrete essentially infinite ideal Ad,∞, the largest type II finite ideal AII,1, the largest type II essentially infinite ideal AII,∞, and the largest type III ideal AIII of any C*-algebra A such that Ad,1 + Ad,∞ + AII,1 + AII,∞ + AIII is an essential ideal of A. This “decomposition” extends the corresponding one for W*-algebras.We also give a closer look at C*-algebras with Hausdorff primitive ideal spaces, AW*-algebras as well as local multiplier algebras of C*-algebras. We find that these algebras can be decomposed into continuous fields of prime C*-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above.