Noncommutative Jordan rings
Kevin
McCrimmon
1-33
Abstract: Heretofore most investigations of noncommutative Jordan algebras have been restricted to algebras over fields of characteristic $\ne 2$ in order to make use of the passage from a noncommutative Jordan algebra $ \mathfrak{A}$ to the commutative Jordan algebra ${\mathfrak{A}^ + }$ with multiplication $x \cdot y = \frac{1}{2}(xy + yx)$. We have recently shown that from an arbitrary noncommutative Jordan algebra $ \mathfrak{A}$ one can construct a quadratic Jordan algebra ${\mathfrak{A}^ + }$ with a multiplication $ {U_x}y = x(xy + yx) - {x^2}y = (xy + yx)x - y{x^2}$, and that these quadratic Jordan algebras have a theory analogous to that of commutative Jordan algebras. In this paper we make use of this passage from $ \mathfrak{A}$ to ${\mathfrak{A}^ + }$ to derive a general structure theory for noncommutative Jordan rings. We define a Jacobson radical and show it coincides with the nil radical for rings with descending chain condition on inner ideals; semisimple rings with d.c.c. are shown to be direct sums of simple rings, and the simple rings to be essentially the familiar ones. In addition we obtain results, which seem to be new even in characteristic $ \ne 2$, concerning algebras without finiteness conditions. We show that an arbitrary simple noncommutative Jordan ring containing two nonzero idempotents whose sum is not 1 is either commutative or quasiassociative.
Isolated invariant sets and isolating blocks
C.
Conley;
R.
Easton
35-61
Structure of the semigroup of semigroup extensions
R. O.
Fulp;
J. W.
Stepp
63-73
Abstract: Let $B$ denote a compact semigroup with identity and $G$ a compact abelian group. Let $\operatorname{Ext} (B,G)$ denote the semigroup of extensions of $G$ by $B$. We show that $\operatorname{Ext} (B,G)$ is always a union of groups. We show that it is a semilattice whenever $ B$ is. In case $ B$ is also an abelian inverse semigroup with its subspace of idempotent elements totally disconnected, we obtain a determination of the maximal groups of a commutative version of $ \operatorname{Ext} (B,G)$ in terms of the extension functor of discrete abelian groups.
Generating and cogenerating structures
John A.
Beachy
75-92
Abstract: A functor $ T:\mathcal{A} \to \mathcal{B}$ acts faithfully on the right of a class of objects $ \mathcal{A}$ if it distinguishes morphisms out of objects of $T(f) \ne T(g))$. We define a full subcategory $ \mathcal{R}\mathcal{F}(T)$ such that $T$ acts faithfully on the right of the objects of $ \mathcal{R}\mathcal{F}(T)$. An object $ U \in \mathcal{A}$ is a generator if ${H^U}:\mathcal{A} \to \mathcal{E}ns$ is faithful, and if ${H^U}$ is not faithful, we may still consider $ \mathcal{R}\mathcal{F}({H^U})$. This gives rise to the notion of a generating structure. Cogenerating structures are defined dually, and various canonical generating and cogenerating structures are defined for the category of $ R$-modules. Relationships between these can be used in the homological classification of rings.
Algebraic models for probability measures associated with stochastic processes
B. M.
Schreiber;
T.-C.
Sun;
A. T.
Bharucha-Reid
93-105
Abstract: This paper initiates the study of probability measures corresponding to stochastic processes based on the Dinculeanu-Foiaş notion of algebraic models for probability measures. The main result is a general extension theorem of Kolmogorov type which can be summarized as follows: Let $\{ (X,{\mathcal{A}_i},{\mu _i}),i \in I\}$ be a directed family of probability measure spaces. Then there is an associated directed family of probability measure spaces $\{ (G,{\mathcal{B}_i},{v_i}),i \in I\}$ and a probability measure $ v$ on the $\sigma $-algebra $\mathcal{B}$ generated by the ${\mathcal{B}_i}$ such that (i) $v(B) = {v_i}(B),B \in {\mathcal{B}_i},i \in I$, and (ii) for each is $i \in I$ the spaces $(X,{\mathcal{A}_i},{\mu _i})$ and $ (G,{\mathcal{B}_i},{v_i})$ are conjugate. The importance of the main theorem is that under certain mild conditions there exists an embedding $ \psi :X \to G$ such that the induced measures ${v_i}$ on $G$ are extendable to $v$, although the measures ${\mu _i}$ on $X$ may not be extendable. Using the algebraic model formulation, the Kolmogorov extension property and the notion of a representation of a directed family of probability measure spaces are discussed.
Cyclic vectors and irreducibility for principal series representations.
Nolan R.
Wallach
107-113
Abstract: Canonical sets of cyclic vectors for principal series representations of semisimple Lie groups having faithful representations are found. These cyclic vectors are used to obtain estimates for the number of irreducible subrepresentations of a principal series representations. The results are used to prove irreducibility for the full principal series of complex semisimple Lie groups and for $SL(2n + 1,R),n \geqq 1$.
Involutory automorphisms of operator algebras
E. B.
Davies
115-142
Abstract: We develop the mathematical machinery necessary in order to describe systematically the commutation and anticommutation relations of the field algebras of an algebraic quantum field theory of the fermion type. In this context it is possible to construct a skew tensor product of two von Neumann algebras and completely describe its type in terms of the types of the constituent algebras. Mathematically the paper is a study of involutory automorphisms of ${W^\ast}$-algebras, of particular importance to quantum field theory being the outer involutory automorphisms of the type III factors. It is shown that each of the hyperfinite type III factors studied by Powers has at least two outer involutory automorphisms not conjugate under the group of all automorphisms of the factor.
The curvature of level curves
Dorothy Browne
Shaffer
143-150
Abstract: Sharp bounds are derived for the curvature of level curves of analytic functions in the complex plane whose logarithmic derivative has the representation $ c/(w - g(w))$, where $ g(w)$ is analytic for $ \vert w\vert > a$ and $\vert g(w)\vert \leqq a,c$ real. These results are applied in particular to lemniscates and sharpened for the level curves of lacunary polynomials. Extensions to the level curves of Green's function and rational functions are indicated.
Noncommutative Jordan algebras of capacity two
Kirby C.
Smith
151-159
Abstract: Let $J$ be a noncommutative Jordan algebra with 1. If $J$ has two orthogonal idempotents $e$ and $f$ such that $1 = e + f$ and such that the Peirce $1$-spaces of each are Jordan division rings, then $J$ is said to have capacity two. We prove that a simple noncommutative Jordan algebra of capacity two is either a Jordan matrix algebra, a quasi-associative algebra, or a type of quadratic algebra whose plus algebra is a Jordan algebra determined by a nondegenerate symmetric bilinear form.
Arcwise connectedness of semiaposyndetic plane continua
Charles L.
Hagopian
161-165
Abstract: In a recent paper, the author proved that if a compact plane continuum $ M$ contains a finite point set $F$ such that, for each point $x$ in $M - F,M$ is semi-locally-connected and not aposyndetic at $x$, then $M$ is arcwise connected. The primary purpose of this paper is to generalize that theorem. Semiaposyndesis, a generalization of semi-local-connectedness, is defined and arcwise connectedness is established for certain semiaposyndetic plane continua.
A priori estimates and unique continuation theorems for second order parabolic equations
Raymond
Johnson
167-177
Abstract: It is shown that solutions of second-order linear parabolic equations subject to global constraint satisfy an a priori estimate which has among its consequences that if a solution of such an equation vanishes on the characteristic $t = T$ and satisfies the global constraint, it vanishes identically.
Regular identities in lattices
R.
Padmanabhan
179-188
Abstract: An algebraic system $\mathfrak{A} = \langle A; + , \circ \rangle$ is called a quasilattice if the two binary operations + and $ ^\circ$ are semilattice operations such that the natural partial order relation determined by + enjoys the substitution property with respect to $^\circ$ and vice versa. An identity ``$f = g$'' in an algebra is called regular if the set of variables occurring in the polynomial $ f$ is the same as that in $ g$. It is called $ n$-ary if the number of variables involved in it is at the most $n$. In this paper we show that the class of all quasilattices is definable by means of ternary regular lattice identities and that these identities span the set of all regular lattice identities and that the arity of these defining equations is the best possible. From these results it is deduced that the class of all quasilattices is the smallest equational class containing both the class of all lattices and the class of all semilattices in the lattice of all equational classes of algebras of type $\langle 2,2\rangle$ and that the lattice of all equational classes of quasilattices is distributive.
On the order of a starlike function
F.
Holland;
D. K.
Thomas
189-201
Abstract: It is shown that if $f \in S$, the class of normalised starlike functions in the unit $\operatorname{disc} \Delta $, then $\displaystyle ({\text{i}})\quad \quad \mathop {\lim }\limits_{r \to 1 - } \frac... ...P_\lambda }(r)}}{{ - \log (1 - r)}} = \alpha \lambda {\text{ for }}\lambda > 0;$ $\displaystyle ({\text{ii}})\quad \quad \mathop {\lim }\limits_{r \to 1 - } \fra... ...}\vert{\vert _p}}}{{ - \log (1 - r)}} = \alpha p - 1{\text{ for }}\alpha p > 1;$ and
On the $C\sp*$-algebra of Toeplitz operators on the quarterplane
R. G.
Douglas;
Roger
Howe
203-217
Abstract: Using the device of the tensor product of $ {C^ \ast }$-algebras, we study the ${C^ \ast }$-algebra generated by the Toeplitz operators on the quarter-plane. We obtain necessary and sufficient conditions for such an operator to be Fredholm, but show in this case that not all such operators are invertible.
Operations in polyadic algebras
Aubert
Daigneault
219-229
Abstract: A new treatment of P. R. Halmos' theory of terms and operations in (locally finite) polyadic algebras (of infinite degree) is given that is considerably simpler than the original one.
Matching theory for combinatorial geometries
Martin
Aigner;
Thomas A.
Dowling
231-245
Abstract: Given two combinatorial (pre-) geometries and an arbitrary binary relation between their point sets, a matching is a subrelation which defines a bijection between independent sets of the geometries. The theory of matchings of maximum cardinality is developed in two directions, one of an algorithmic, the other of a structural nature. In the first part, the concept of an augmenting chain is introduced to establish as principal results a min-max type theorem and a generalized Marriage Theorem. In the second part, Ore's notion of a deficiency function for bipartite graphs is extended to determine the structure of the set of critical sets, i.e. those with maximum deficiency. The two parts of the investigation are then connected using the theory of Galois connections.
On the classification of symmetric graphs with a prime number of vertices
Chong-yun
Chao
247-256
Abstract: We determine all the symmetric graphs with a prime number of vertices. We also determine the structure of their groups.