We study non-commutative convolution measure algebras satisfying the condition in the title and having an involution with a non-degenerate finite dimensional *-representation. We show first that the group algebra L1(G) of a locally compact group G satisfies these conditions. Then we show that to a given algebra A with the above conditions there corresponds a locally compact group G such that A is a * and L-subalgebra of M(G) and such that the enveloping C*-algebra of A is *isomorphic to C*(G). Finally we show for certain groups that L1(G) is the only example of such algebras, thus giving a characterisation of L1(G).

Let p be a prime number and let k be a field of characteristic not equal to p. Assuming k contains the appropriate roots of unity, we characterise the non-cyclic Galois extensions of k of degree p3. Concrete examples of such extensions are given for each possible case which can occur, up to isomorphism.

A graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma $ and the set of s-arcs of $\Gamma $ , where for an integer $s \ge 1$ an s-arc of $\Gamma $ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma $ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$ . A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$ -arc-transitive but not $(\text{Aut} (\Gamma ), 3)$ -arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if $\Gamma $ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either $\Gamma $ is normal or G is one of the groups $\text{PSL}_2(11)$ , ${\rm M} _{11}$ , $\text{M} _{23}$ and $A_{11}$ . However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

Let A, B ⊆ {1, …, n}. For m ∈ Z, let rA,B(m) be the cardinality of the set of ordered pairs (a, b) ∈ A × B such that a + b = m. For t ≥ 1, denote by (A + B)t the set of the elements m for which rA, B(m) ≥ t. In this paper we prove that for any subsets A, B ⊆ {1, …, n} such that |A| + |B| ≥ (4n + 4t − 3)/3, the sumset (A + B)t contains a block of consecutive integers with the length at least |A| + |B| − 2t + 1, and that (a) for any two subsets A and B of {1, …, n} such that |A| + |B| ≥ (4n)/3, there exists an arithmetic progression of length n in A + B; (b) for any 2 ≤ r ≤ (4n − 1)/3, there exist two subsets A and B of {1, …, n} with |A| + |B| = r such that any arithmetic progression in A + B has the length at most (2n − 1)/3 + 1.

If μ and γ are ordinal numbers such that for every α ≤ γ·2, then γ < ω. This localizes a result due to Patai.

By extending the given action of a discrete group G on a Tychonoff space X to βX, it is proved that the Stone-Čech compactification of the orbit space of X is the orbit space of the Stone-Čech compactification βX of X, when G is finite. The notion of G-retractive spaces is introduced and it is proved that the orbit space of a G-retractive space with G finite, is G-retractive.

D. Gromoll and K. Grove showed that metric flows on constant curvature spaces are either flat or locally spanned by Killing vector fields. We generalise this result to certain flows on manifolds of variable curvature.

A Skorokhod embedding approach is used to give the other law of the iterated logarithm for square integrable martingales.

In Theorem 3.5 ([1], p. 407), we need the base space X to be compact Hausdorff, so that it has a finite partition of unity.

Generalising the concept of injectivity, we study the notion of $\varphi $ -injectivity for dual Banach algebras. It provides a framework for studying $\varphi $ -amenability of enveloping dual Banach algebras.

We use Ramsey's theorem to generalise a result of L. Babai and T.S. Sós on Sidon subsets and then use this to prove that for an integer n > 1 the class of groups in which every infinite subset contains a rewritable n-subset coincides with the class of groups in which ever infinite subset contains n mutually disjoint non-empty subsets X1,…,Xn such that X1…Xn∩Xσ(1) …xσ(n) ≠ θ for some non-identity permutation σ on the set {1,…,n}.

We prove that the Fridman invariant defined using the Carathéodory pseudodistance does not always go to 1 near strongly Levi pseudoconvex boundary points and it always goes to 0 near nonpseudoconvex boundary points. We also discuss whether Fridman invariants can be extended continuously to some boundary points of domains constructed by deleting compact subsets from other domains.

Let G be a finite solvable group and let p be a prime divisor of $|G|$ . We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.

We show that the left joint spectrum of an arbitrary n-tuple of hyponormal Hilbert space operators can be obtained from the spectral set γ introduced by McIntosh and Pryde. A dual statement for cohyponormal operators is also true. The result is a generalisation of a theorem proved by Pryde and the author for normal operators.

The main result of this paper is that there exist non-principal left ideals in a certain twisted group algebra A of the infinite dihedral group ≤ a, b | b−1ab = a−1, b2 = 1 ≥ over the field R of real numbers: namely in the A defined by b−l ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ.

The motivation comes from the study (in a series of papers by Berman and the author) of finitely generated torsion-free RG-modules for groups G which have an infinite cyclic subgroup of finite index. In a sense, this amounts to studying modules over (full matrix algebras over) a finite set of R-algebras [namely, for the groups in question, these algebras take on the role played by R, C and H (the real quaternions) in the theory of real representations of finite groups]. For all but two algebras in that finite set, satisfying results have been obtained by exploiting the fact that each of them is either a ring with zero divisors or a principal left ideal ring. The other two are known to have no zero divisors. One of them is the present A. The point of the main result is that new ideas will be needed for understanding A-modules.

A number of subsidiary results are concerned with convenient generating sets for left ideals in A.

Let $\alpha $ be a totally positive algebraic integer of degree d, with conjugates $\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$ . The absolute $S_k$ -measure of $\alpha $ is defined by $s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$ . We compute the lower bounds $\upsilon _k$ of $s_k(\alpha )$ for each integer in the range $2\leq k \leq 15$ and give a conjecture on the results for integers $k>15$ . Then we derive the lower bounds of $s_k(\alpha )$ for all real numbers $k>2$ . Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.