Decomposition rank and \mathcal{Z} -stability

We show that separable, simple, nonelementary, unital C^*-algebras with finite decomposition rank absorb the Jiang–Su algebra Z\mathcal{Z} tensorially. This has a number of consequences for Elliott’s program to classify nuclear C^*-algebras by their K-theory data. In particular, it completes the classification of C^*-algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K-groups.     

Recasting the Elliott conjecture

Let A be a simple, unital, finite, and exact C^*-algebra which absorbs the Jiang–Su algebra tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases— -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of -stable C^*-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that -theoretic invariants will classify separable and nuclear C^*-algebras—with the recent appearance of counterexamples to its strongest concrete form. Research supported by the DGI MEC-FEDER through Project MTM2005-00934, and the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. A. S. Toms was also supported in part by an NSERC Discovery Grant.     

Maximum Hitting of a Set by Compressed Intersecting Families

For a family A{\mathcal{A}} and a set Z, denote {A ? A \colon A ?Z 1 ?}{\{A \in \mathcal{A} \colon A \cap Z \neq \emptyset\}} by A(Z){\mathcal{A}(Z)}. For positive integers n and r, let S_n,r{\mathcal{S}_{n,r}} be the trivial compressed intersecting family {A ? (c[n]r ) \colon 1 ? A}{\{A \in \big(\begin{subarray}{c}[n]\\r \end{subarray}\big) \colon 1 \in A\}}, where [n] : = {1, ?, n}{[n] := \{1, \ldots, n\}} and (c[n]r ) : = {A ì [n] \colon |A| = r}{\big(\begin{subarray}{c}[n]\\r \end{subarray}\big) := \{A \subset [n] \colon |A| = r\}}. The following problem is considered: For r ≤ n/2, which sets Z í [n]{Z \subseteq [n]} have the property that |A(Z)| ￡ |S_n,r(Z)|{|\mathcal{A}(Z)| \leq |\mathcal{S}_{n,r}(Z)|} for any compressed intersecting family A ì (c[n]r ){\mathcal{A}\subset \big(\begin{subarray}{c}[n]\\r \end{subarray}\big)}? (The answer for the case 1 ? Z{1 \in Z} is given by the Erdős–Ko–Rado Theorem.) We give a complete answer for the case |Z| ≥ r and a partial answer for the much harder case |Z| < r. This paper is motivated by the observation that certain interesting results in extremal set theory can be proved by answering the question above for particular sets Z. Using our result for the special case when Z is the r-segment {2, ?, r+1}{\{2, \ldots, r+1\}}, we obtain new short proofs of two well-known Hilton–Milner theorems. At the other extreme end, by establishing that |A(Z)| ￡ |S_n,r(Z)|{|\mathcal{A}(Z)| \leq |\mathcal{S}_{n,r}(Z)|} when Z is a final segment, we provide a new short proof of a Holroyd–Talbot extension of the Erdős-Ko-Rado Theorem.     

Asymptotic unitary equivalence and classification of simple amenable <Emphasis Type="Italic">C</Emphasis><Superscript>∗</Superscript>-algebras

Let C and A be two unital separable amenable simple C ^?-algebras with tracial rank at most one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ? ₁,? ₂:C→A are two unital monomorphisms. We show that there is a continuous path of unitaries {u _t :t∈[0,∞)} of A such that

$\lim_{t\to\infty}u_t^*\varphi_1(c)u_t=\varphi_2(c)\quad\mbox{for all }c\in C$

if and only if [? ₁]=[? ₂] in \(KK(C,A),\varphi_{1}^{\ddag}=\varphi_{2}^{\ddag},(\varphi_{1})_{T}=(\varphi _{2})_{T}\) and a rotation related map \(\overline{R}_{\varphi_{1},\varphi_{2}}\) associated with ? ₁ and ? ₂ is zero.

Applying this result together with a result of W. Winter, we give a classification theorem for a class \({\mathcal{A}}\) of unital separable simple amenable C ^?-algebras which is strictly larger than the class of separable C ^?-algebras with tracial rank zero or one. Tensor products of two C ^?-algebras in \({\mathcal{A}}\) are again in \({\mathcal{A}}\). Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras for which the state space of K ₀ is the same as the tracial state space and also some unital simple ASH-algebras whose K ₀-group is ? and whose tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are \({\mathcal{Z}}\)-stable are isomorphic to ones with no dimension growth.     

On artin algebras with almost all indecomposable modules of projective or injective dimension at most one

Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with co-finite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which is co-finite in ind A, and derive some consequences. Dedicated to Stanislaw Balcerzyk on the occation of his 70th birthday     

Periodicity of Non-Central Integral Arrangements Modulo Positive Integers

An integral coefficient matrix determines an integral arrangement of hyperplanes in \mathbbR^m{\mathbb{R}^m} . After modulo q reduction ${(q \in {\mathbb{Z}_{ >0 }})}${(q \in {\mathbb{Z}_{ >0 }})} , the same matrix determines an arrangement A_q{\mathcal{A}_q} of “hyperplanes” in \mathbbZ^m_q{\mathbb{Z}^m_q} . In the special case of central arrangements, Kamiya, Takemura, and Terao [J. Algebraic Combin. 27(3), 317–330 (2008)] showed that the cardinality of the complement of A_q{\mathcal{A}_q} in \mathbbZ^m_q{\mathbb{Z}^m_q} is a quasi-polynomial in ${q \in {\mathbb{Z}_{ >0 }}}${q \in {\mathbb{Z}_{ >0 }}} . Moreover, they proved in the central case that the intersection lattice of A_q{\mathcal{A}_q} is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement [^(B)]_m^[0,a]{\hat{\mathcal{B}}_m^{[0,a]}} of Athanasiadis [J. Algebraic Combin. 10(3), 207–225 (1999)] to illustrate our results.     

Imaginaries in Boolean algebras

Given an infinite Boolean algebra B, we find a natural class of $\varnothing$‐definable equivalence relations $\mathcal {E}_{B}$ such that every imaginary element from B^eq is interdefinable with an element from a sort determined by some equivalence relation from $\mathcal {E}_{B}$. It follows that B together with the family of sorts determined by $\mathcal {E}_{B}$ admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of imaginaries for Boolean algebras, obtained in 10 .     

M?bius transform, moment-angle complexes and Halperin–Carlsson conjecture

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the M?bius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z_K^{(\mathbb D, \mathbb S)}\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex Z_K\mathcal{Z}_{K} and the real moment-angle complex \mathbbRZ_K\mathbb{R}\mathcal {Z}_{K} as special examples. We show that H^*(Z_K^{(\mathbb D, \mathbb S)};k)H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor  ^k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for Z_K\mathcal{Z}_{K} (resp. \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T ^m -action on Z_K\mathcal{Z}_{K} (resp. (ℤ₂) ^m -action on \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}).     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

Multisorted dualisability: change of base

We prove that if a quasivariety A{\mathcal{A}} generated by a finite family M{\mathcal{M}} of finite algebras has a multisorted duality based on M{\mathcal{M}}, then A{\mathcal{A}} has a multisorted duality based on any finite family of finite algebras that generates it.     

On Semisimple Hopf Algebras of Dimension 2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>, II

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension 2²ⁿ⁺¹ for n ≥ 2 over an algebraically closed field of characteristic 0 with the group of group-like elements isomorphic to \(\mathbb {Z}_{2^{n}}\times \mathbb {Z}_{2^{n}}\). Moreover we classify all such nonisomorphic Hopf algebras of dimension 32 and show that they are not twist-equivalent to each other. More generally, given an abelian group of order 2 ^m?1 we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension 2 ^m which have this group as their group of group-like elements.     

Covering and packing in {\mathbb Z^n} and {\mathbb R^n}. (II)

In the present part (II) we will deal with the group \mathbb G = \mathbb Zⁿ{\mathbb G = \mathbb Z^n} , and we will study the effect of linear transformations on minimal covering and maximal packing densities of finite sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} . As a consequence, we will be able to show that the set of all densities for sets A{\mathcal A} of given cardinality is closed, and to characterize four-element sets A ì \mathbb Zⁿ{\mathcal A \subset {\mathbb Z}^n} which are “tiles”. The present work will be largely independent of the first part (I) presented in [4].     

K0 of multiplier algebras of C*-algebras with real rank zero

For a large class of -unital C ^*-algebras A with real rank zero and stable rank one, the structure of the Grothendieck group k ₀ of the multiplier algebra (A) is investigated. The ordered group K ₀( (A)) is shown to be an unperforated Riesz group, and its additive structure is completely determined, as is — in important cases — its order structure. These structures, and the attendant consequences for the ideal structure of (A), are richer than previously anticipated. In particular, it is shown that the corona algebra (A)/A can have very large stably finite quotient algebras. For example, there exist simple, separable, approximately finite-dimensional C ^*-algebras A such that the maximal stably finite quotient algebra of (A)/A has uncountably many maximal ideals modulo which a W ^*-factor of Type II₁ results. The analysis of the additive structure of K ₀( (A)) yields as a byproduct that if A is a -unital approximately finite-dimensional C ^*-algebra without nonzero unital quotient algebras, then all quasitraces on (A) are traces.This research was partially supported by a grant from the National Science Foundation.     

Nilpotent algebras and affinely homogeneous surfaces

To every nilpotent commutative algebra N{\mathcal{N}} of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety S ì N{S\subset\mathcal{N}} can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator A{\mathcal{A}} of N{\mathcal{N}}. In case N{\mathcal{N}} admits a grading, the surface S is affinely homogeneous. More can be said if A{\mathcal{A}} has dimension 1, that is, if N{\mathcal{N}} is the maximal ideal of a Gorenstein algebra. In this case two such algebras N{\mathcal{N}}, [(N)\tilde]{\tilde{\mathcal{N}}} are isomorphic if and only if the associated hypersurfaces S, [(S)\tilde]{\tilde S} are affinely equivalent. If one of S, [(S)\tilde]{\tilde S} even is affinely homogeneous, ‘affinely equivalent’ can be replaced by ‘linearly equivalent’. In case the nil-index of N{\mathcal{N}} does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.     

The Elliott conjecture for Villadsen algebras of the first type

We study the class of simple C^∗-algebras introduced by Villadsen in his pioneering work on perforated ordered K-theory. We establish six equivalent characterisations of the proper subclass which satisfies the strong form of Elliott's classification conjecture: two C^∗-algebraic (Z-stability and approximate divisibility), one K-theoretic (strict comparison of positive elements), and three topological (finite decomposition rank, slow dimension growth, and bounded dimension growth). The equivalence of Z-stability and strict comparison constitutes a stably finite version of Kirchberg's characterisation of purely infinite C^∗-algebras. The other equivalences confirm, for Villadsen's algebras, heretofore conjectural relationships between various notions of good behaviour for nuclear C^∗-algebras.     

Noncommutative Positivstellensätze for pairs representation-vector

We study non-commutative real algebraic geometry for a unital associative *-algebra A{\mathcal {A}} viewing the points as pairs (π, v) where π is an unbounded *-representation of A{\mathcal A} on an inner product space which contains the vector v. We first consider the *-algebras of matrices of usual and free real multivariate polynomials with their natural subsets of points. If all points are allowed then we can obtain results for general A{\mathcal {A}}. Finally, we compare our results with their analogues in the usual (i.e. Schmüdgen’s) non-commutative real algebraic geometry where the points are unbounded *-representation of A{\mathcal {A}}.     

De Morgan Heyting algebras satisfying the identity xn(′*) ≈ x(n+1)(′*)

In this paper we investigate the sequence of subvarieties $ {\mathcal {SDH}_n} $of De Morgan Heyting algebras characterized by the identity x^n(′*) ≈ x^(n+1)(′*). We obtain necessary and sufficient conditions for a De Morgan Heyting algebra to be in $ {\mathcal {SDH}_1} $ by means of its space of prime filters, and we characterize subdirectly irreducible and simple algebras in $ {\mathcal {SDH}_1} $. We extend these results for finite algebras in the general case $ {\mathcal {SDH}_n} $. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Characterizations of Bent and Almost Bent Function on {\mathbb{Z}_p^2}

Bent and almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} in two classes of M{\mathcal{M}} ’s and PS{\mathcal{PS}} ’s, and show that the graph set corresponding to a bent function on \mathbbZ_p²{\mathbb{Z}_p^2} can be written as the sum of a graph set of M{\mathcal{M}} ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of M{\mathcal{M}} ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.     

On Lie derivations of Lie ideals of prime algebras

LetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureA _c and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let be the factor Lie algebra and let δ: be a Lie derivation. Suppose that char(A) ≠ 2 andA does not satisfySt ₁₄, the standard identity of degree 14. We show thatR ΩC =Z(R) and there exists a derivation of algebrasD:A →A _c such that for allx∈R. Our result solves an old problem of Herstein.     

On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime

Given a finite family F\mathcal{F} of linear forms with integer coefficients, and a compact abelian group G, an F\mathcal{F}-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F\mathcal{F}. We denote by d_F(G)d_{\mathcal{F}}(G) the supremum of μ(A) over F\mathcal{F}-free sets A⊂G, where μ is the normalized Haar measure on G. Our main result is that, for any such collection F\mathcal{F} of forms in at least three variables, the sequence d_F(\mathbb Z_p)d_{\mathcal{F}}({\mathbb {Z}}_{p}) converges to d_F(\mathbb R/\mathbb Z)d_{\mathcal{F}}({\mathbb {R}}/{\mathbb {Z}}) as p→∞ over primes. This answers an analogue for ℤ _p of a question that Ruzsa raised about sets of integers.