Classification of Homomorphisms from C (X) to Simple C*-Algebras of Real Rank Zero

Abstract Let A be a unital simple C*-algebra of real zero, stable rank one, with weakly unperforated K ₀₍ A) and unique normalized quasi-trace τ, and let X be a compact metric space. We show that two monomorphisms φ, ψ : C(X)→A are approximately unitarily equivalent if and only if φ and ψ induce the same element in KL(C(X), A) and the two lineal functionals τ∘φ and τ∘ψ are equal. We also show that, with an injectivity condition, an almost multiplicative morphism from C(X) into A with vanishing KK-obstacle is close to a homomorphism. Research partially supported by NSF Grants DMS 93-01082 (H.L) and DMS-9401515(G.G). This work was reported by the first named author at West Coast Operator Algebras Seminar (Sept. 1995, Eugene, Oregon)     

Estimation of the number of fixed points of map extensions

Letf:(X,A)→(X,A) be an extension of a given map ψ:A→A, where (X,A) is a pair of compact polyhedra. We shall introduce a special Nielsen number,SN(f|ψ), which is a lower bound for the number of fixed points onX-A for all extensions in the homotopy class off. It is shown that for many space pairs this lower bound is the best possible one, and that it can be realized without the by-passing condition.     

The equilibrium states for semigroups of rational maps

We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψ and a unique exp(P(ψ) − ψ)-conformal measure, where P(ψ) denotes the topological pressure of ψ. The research of M. Urbański was supported in part by the NSF Grant DMS 0400481. H. Sumi thanks University of North Texas for kind hospitality, during his stay there.     

Projective modules in classical and quantum functional analysis

Along with the classical version, there are two “quantized” versions of the theory of operator algebras. In these lectures, the fundamental homological notion of a projective module is described in the framework of these three theories. Our initial definitions of projectivity do not go far from their prototypes in abstract algebra; however, the principal results concern essentially functional-analytic objects and, as a rule, have no purely algebraic analogs. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 7, pp. 7–84, 2007.     

ψ-correct decision for selection and elimination

Summary The selection oft out ofk populations with parameters θ _i (i=1, ...,k) is said to result in an ψ-correct decision provided ψ (minimum selected θ)>maximum non-selected θ where ψ(θ) (>θ) is an increasing function. For the cases of location or scale parameters the minimum probability of ψ-correct decision over the entire parameter space is shown to be no less than the minimum probability of correct selection over a preference zone determined by ψ(θ). For other types of parameters this result is shown to be true under certain conditions linking the distribution function and the ψ function.     

On subadditive functions and ψ-additive mappings

In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive     

A comparison of the method of separation of physical variables with the two-field method for modeling the fluid dynamics of casting

We use the example of the two-dimensional problem of siphon filling of a mold by molten metal to compare two methods of studying the flow of a viscous fluid: in vortex-function flow variables (ω, ψ) and in pressurevelocity variables (p, v). The superiority of the (ω, ψ)-method is demonstrated. Two figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 130–132.     

Sharp forms of nevanlinna’s error terms

Let f(z) be a meromorphic function in the plane. If ψ(t)/t andp(t) are two positive, continuous and non-decreasing functions on [1,∞) with ∫ ₁ ^∞ dt/ψ(t) = ∞ and ∫ ₁ ^∞ dt/p(t) = ∞, then asr → ∞ outside a small exceptional set, provided that the divergence of the integral ∫ ₁ ^r dt/ψ(t) is slow enough. The same forms for the logarithmic derivative and for the ramification term are obtained. It is shown by example that the estimates are best possible. Author supported by Max-Planck-Gesellschaft Z.F.D.W and by NSFC.     

The reconstruction theorem for endomorphisms

The reconstruction theorem deals with dynamical systems which are given by a map ψ : M → M together with a read out function 𝒻 : M → ℝ. Restricting to the cases where ψ is a diffeomorphism, it states that for generic (ψ, 𝒻 ) there is a bijection between elements x ∈ M and corresponding sequences (𝒻(x), 𝒻 (ψ(x)), . . . , 𝒻 (ψ ^{k -1}(x))) of k successive observations, at least for k sufficiently big. This statement turns out to be wrong in cases where ψ is an endomorphism. In the present paper we derive a version of this theorem for endomorphisms (and which is equivalent to the original theorem in the case of diffeomorphisms). It justifies, also for dynamical systems given by endomorphisms, the algorithms for estimating dimensions and entropies of attractors from obervations. Received: 20 June 2002     

A generalization of the rees theorem to a class of regular semigroups

Let S be a regular semigroup for which Green's relations J and D coincide, and which is max-principal in the sense that every element of S is contained in maximal principal right, left and two-sided ideals of S. A construction is given of a max-principal regular semigroup W with J=D, which is also principally separated in the sense that distinct maximal principal right (or left) ideals of S are disjoint, and an epimorphism ψ: W→S that preserves maximality of principal left, right, and two sided ideals, and is in a sense locally one-to-one. If S is completely simple, this construction reduces to the Rees matrix representation of S. The main result of this paper has its origin in an incorrect result contained in the author's doctoral dissertation which was written at the University of California (Berkeley) under Professor John Rhodes. This theorem was first established for finite regular semigroups in [1] (Corollary 2.3), and the present generalization of this result to infinite semigroups was suggested by Professor A. H. Clifford, who the author would like to thank for this as well as his generous encouragement and many helpful editorial suggestions. The author would also like to thank Professor Rhodes for his encouragement.     

Provability interpretations of modal logic

We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev ^* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.     

Pseudo-Eisenstein forms and cohomology¶of arithmetic groups. I

Let ψ be a compactly supported closed differential form on the e[P] of the Borel–Serre boundary of an arithmetically defined locally symmetric space S. A closed compactly supported differential form E (ψ) on S is defined by a pseudo-Eisenstein series attached to ψ. Its degree is the degree of ψ shifted by the codimension of e[P] in S. Non-vanishing results for the cohomology class [E(ψ)] represented by E(ψ) are obtained by use of Poincaré duality and results on cohomology classes represented by ordinary Eisenstein series. Received: 21 July 2001 / Revised version: 17 September 2001     

When is a Markovian Semigroup Induced by a Semiflow?

t , for t ≥ 0, be a strongly continuous Markovian semigroup acting on C(X), where X is a compact Hausdorf space, and let D denote the domain of its infinitesimal generator Z. Suppose D contains a (perhaps finite) family of functions f separating the points of X and satisfying Zf² = 2fZf. If either (1) there exists δ > 0 such that (T^t f)²∈ D if 0 ≤ t ≤δ for each f in this family; or (1′) for some core D′ of Z, g ∈ D′ implies g²∈ D, then the underlying Markoff process on X is deterministic. That is, there exists a semiflow — a semigroup (under composition) of continuous functions φ_t from X into X — such that T^tf(x) = f(φ_t (x)). If the domain D should be an algebra then conditions (1) and (1′) hold trivially. Conversely, if we have a separating family satisfying Zf² = 2fZf then each of these conditions implies that D is an algebra. It is an open question as to whether these conditions are redundant. If the functions φ_t are homeomorphisms from X onto X, then of course we have a Markovian group induced by a flow. This result is obtained by first providing general results about the null-space N of the (function-valued) positive semidefinite quadratic form defined by < f, g > = Z(fg) - fZg - gZf. The set N can be defined for any generator Z of a strongly continuous Markovian semigroup and is equivalently given by N = {f ∈ D| f²∈ D and Zf² = 2fZf} = {f ∈ D| T^t(f²)-(T^tf)² is o(t²) in C(X)}. In the general case N is an algebra closed under composition with any C¹-function φ from the reals to the reals, and Z(φ[f]) = (Zf)φ′[f] if f ∈ N. This "chain rule" on N (on which Z must act as a derivation) is a special case of a theorem for C²-functions φ which holds more generally for all f in d, viz., Z(φ[f] = (Zf) φ′[f] + ? <f, f> φ″[f], Provided Z is a local operator and D is an algebra. In this case the form < f, g > itself enjoys the relation < φ[f], ψ[g] > = φ′ [f] ψ′[g] < f, g >, for C²functions φ and ψ. Some of the results and their proofs continue to hold when the setting is switched from the commutative C*-algebra C(X) to a general (noncommutative) C*-algebra A. In the norm continuous case we obtain a sharp characterization of Markovian semigroups that are groups: Let T^t = e^tz , defined for t ≥ 0, be a Markovian semigroup acting on a C*-algebra A that is norm continuous, i.e., ||T^t - I|| ⇒ 0 as t ⇒ 0 +. Assume Z(a²) = a(Za) + (Za) a for some (perhaps finite) set of self-adjoint elements a that generate a Jordan algebra dense among the self-adjoint elements of A. The e^tz , -∞ < t < ∞, is a group of Markovian operators.     

Criteria for analyticity of subordinate semigroups

Let ψ be a Bernstein function. A. Carasso and T. Kato obtained necessary and sufficient conditions for ψ to have the property that ψ(A) generates a quasibounded holomorphic semigroup for every generator A of a bounded C ₀-semigroup in a Banach space, in terms of some convolution semigroup of measures associated with ψ. We give an alternative to Carasso-Kato’s criterion, and derive several sufficient conditions for ψ to have the above-mentioned property. The author was supported in part by the State Program of Fundamental Research of Republic of Belarus under the contract number 20061473.     

Characters,genericity, and multiplicity one for U(3)

Let ψ:U→ℂ be a generic character of the unipotent radicalU of a Borel subgroup of a quasisplitp-adic groupG. The number (0 or 1) of ψ-Whittaker models on an admissible irreducible representation π ofG was expressed by Rodier in terms of the limit of values of the trace of π at certain measures concentrated near the origin. An analogous statement holds in the twisted case. This twisted analogue is used in [F, p. 47] to provide a local proof of the multiplicity one theorem for U(3). This assert that each discrete spectrum automorphic representation of the quasisplit unitary group U(3) associated with a quadratic extensionE/F of number fields occurs in the discrete spectrum with multiplicity one. It is pointed out in [F, p. 47] that a proof of the twisted analogue of Rodier's theorem does not appear in print. It is then given below. Detailing this proof is necessitated in particular by the fact that the attempt in [F, p. 48] at a global proof of the multiplicity one theorem for U(3), although widely quoted, is incomplete, as we point out here. Partially supported by a Lady Davis Visiting Professorship at the Hebrew University and the Max-Planck-Institut für Mathematik, Bonn.     

On Toeplitz operators similar to unilateral shifts

Let T be a Toeplitz operator on the Hardy space H² on the unit circle, and let the symbol of T be of the form ϕ/ψ, where ϕ is an inner function, ψ is a finite Blaschke product, and deg ψ ≤ deg ϕ. D. N. Clark proved that such an operator T is similar to an isometry. In this paper, we find necessary and sufficient conditions under which such an operator T is similar to a unilateral shift. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 85–104.     

A characterization of alternating groups. II

Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every two non-commuting members of X generate a subgroup isomorphic to A₄ or to A₅. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they generate a subgroup isomorphic to A₄. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X) and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A₅ then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2) The subgroup 〈X^G〉 is a direct product of subgroups 〈C _α〉-generated by some connected components C _α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup isomorphic to A₅. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A₅ or is cyclic of order 5. Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006.     

Ψ-density topology

We study the notion of ψ-density point and ψ-density topologies for non-decreasing continuous functions ψ. Also basic properties of ψ-approximately continuous functions are considered.