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Inspired by recent developments in the theory of Banach and operator algebras of locally compact groups, we construct several new classes of bifunctors (A,B)\mapsto A\otimes _{\alpha } B, where A\otimes _\alpha B is a cross norm completion of A\odot B for each pair of C*-algebras A and B. For the first class of bifunctors considered (A,B)\mapsto A{\otimes _p} B (1\leq p\leq \infty), A{\otimes _p} B is a Banach algebra cross-norm completion of A\odot B constructed in a fashion similar to p-pseudofunctions \text {PF}^*_p(G) of a locally compact group. Taking a cue from the recently introduced symmetrized p-pseudofunctions due to Liao and Yu and later by the second and the third named authors, we also consider {\otimes _{p,q}} for Hölder conjugate p,q\in [1,\infty ] – a Banach *-algebra analogue of the tensor product {\otimes _{p,q}}. By taking enveloping C*-algebras of A{\otimes _{p,q}} B, we arrive at a third bifunctor (A,B)\mapsto A{\otimes _{\mathrm C^*_{p,q}}} B where the resulting algebra A{\otimes _{\mathrm C^*_{p,q}}} B is a C*-algebra. For G_1 and G_2 belonging to a large class of discrete groups, we show that the tensor products \mathrm C^*_{\mathrm r}(G_1){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G_2) coincide with a Brown-Guentner type C*-completion of \mathrm \ell ^1(G_1\times G_2) and conclude that if 2\leq p'

In this paper, we characterize closed ideals with bounded approximate identities in some Banach algebras. Our results provide answers to a question of Lau and Ülger. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer. [ABSTRACT FROM AUTHOR]

A remarkable result of Molnár [Proc. Amer. Math. Soc. 126 (1998), pp. 853–861] states that automorphisms of the algebra of operators acting on a separable Hilbert space are stable under "small" perturbations. More precisely, if \phi,\psi are endomorphisms of \mathcal {B}(\mathcal {H}) such that \|\phi (A)-\psi (A)\|<\|A\| and \psi is surjective, then so is \phi. The aim of this paper is to extend this result to a larger class of Banach spaces including \ell _p and L_p spaces, where 1

Spin^h manifolds are the quaternionic analogue to \text {spin}^c manifolds. We compute the \text {spin}^h bordism groups at the prime 2 by proving a structure theorem for the cohomology of the \text {spin}^h bordism spectrum \mathrm {MSpin^h} as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of \mathrm {MSpin^h} as a wedge sum of familiar spectra. We also compute the decomposition of H^*(\mathrm {MSpin^h};\mathbb {Z}/2\mathbb {Z}) explicitly in degrees up through 30 via a counting process. [ABSTRACT FROM AUTHOR]

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let \mathcal {G} be a topological group, and \mathcal {A} a unital symmetric C^*-subalgebra of \mathrm {UC}(\mathcal {G}), the algebra of bounded uniformly continuous functions on \mathcal {G}. Generalizing the notion of a stable metric, we study \mathcal {A}-metrics \delta, i.e., the function \delta (e, \cdot) belongs to \mathcal {A}; the case \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}), the algebra of weakly almost periodic functions on \mathcal {G}, recovers stability. If the topology of G is induced by a left invariant metric d, we prove that \mathcal {A} determines the topology of \mathcal {G} if and only if d is uniformly equivalent to a left invariant \mathcal {A}-metric. As an application, we show that the additive group of C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now \mathcal {G} be a metric group, and assume \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}), the algebra of bounded left uniformly continuous functions on \mathcal {G}, is a unital C^*-algebra which is the uniform closure of coefficients of representations of \mathcal {G} on members of \mathscr {F}, where \mathscr {F} is a class of Banach spaces closed under \ell _2-direct sums. We prove that \mathcal {A} determines the topology of \mathcal {G} if and only if \mathcal {G} embeds into the isometry group of a member of \mathscr {F}, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability. [ABSTRACT FROM AUTHOR]

We exhibit a Banach space Z failing the approximation property, for which there is an uncountable family \mathscr F of closed subideals contained in the Banach algebra \mathcal K(Z) of the compact operators on Z, such that the subideals in \mathscr F are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras \mathcal L(X) of bounded operators on X, where closed ideals \mathcal I \neq \mathcal J are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals contained in the strictly singular operators \mathcal S(X) for classical spaces such as X = L^p with p \neq 2, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur. [ABSTRACT FROM AUTHOR]

Let A be a Banach algebra, X a closed subspace of A*, Y a dual Banach space with predual Y*, and π a continuous representation of A on Y. We call π subordinate to X if each coordinate function πy,λ ∊ X, for all y ∊ Y, λ ∊ Y*. If X is topologically left (right) introverted and Y is reflexive, we show the existence of a natural bijection between continuous representations of A on Y subordinate to X, and normal representations of X* on Y. We show that if A has a bounded approximate identity, then every weakly almost periodic functional on A is a coordinate function of a continuous representation of A subordinate to WAP(A). We show that a function f on a locally compact group G is left uniformly continuous if and only if it is the coordinate function of the conjugate representation of L1(G), associated to some unitary representation of G. We generalize the latter result to an arbitrary Banach algebra with bounded right approximate identity. We prove the functionals in LUC(A) are all coordinate functions of some norm continuous representation of A on a dual Banach space. [ABSTRACT FROM AUTHOR]

We show that for any separable reflexive Banach space X and a large class of Banach spaces E, including those with a subsymmetric shrinking basis but also all spaces L_p[0,1] for 1\le p \le \infty, every bounded linear map \mathcal {B}(E)\to \mathcal {B}(X) which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism \mathcal {B}(E)\to \mathcal {B}(X). That is, the pair (\mathcal {B}(E),\mathcal {B}(X)) has the AMNM property in the sense of Johnson [J. London Math. Soc. (2) 37 (1988), pp. 294–316]. Previously this was only known for E=X=\ell _p with 1

We initiate the large scale geometric study of Banach-Lie groups, especially of linear Banach-Lie groups. We show that the exponential length, originally introduced by Ringrose for unitary groups of C^*-algebras, defines the quasi-isometry type of any connected Banach-Lie group. As an illustrative example, we consider unitary groups of separable abelian unital C^*-algebras with spectrum having finitely many components, which we classify up to topological isomorphism and up to quasi-isometry, in order to highlight the difference. The main results then concern the Haagerup property, and Properties (T) and (FH). We present the first non-trivial non-abelian and non-locally compact groups having the Haagerup property, most of them being non-amenable. These are the groups \mathcal {U}_2(M,\tau), where M is a semifinite von Neumann algebra with a normal faithful semifinite trace \tau. Finally, we investigate the groups \operatorname {E}_n(A), which are closed subgroups of \operatorname {GL}(n,A) generated by elementary matrices, where A is a unital Banach algebra. We show that for n\geq 3, all these groups have Property (T) and they are unbounded, so they have Property (FH) non-trivially. On the other hand, if A is an infinite-dimensional unital C^*-algebra, then \operatorname {E}_2(A) does not have the Haagerup property. If A is moreover abelian and separable, then \operatorname {SL}(2,A) does not have the Haagerup property. [ABSTRACT FROM AUTHOR]

Let \omega :\mathbb Z^2 \to [1,\infty) be a weight satisfying a certain technical property. Let f be a continuous function on the torus \mathbb T^2 such that \sum _{\mathbf {m}\in \mathbb Z^2}|\widehat f(\mathbf {m})|\omega (\mathbf {m})<\infty. If f is nowhere vanishing on \mathbb T^2, then there is a weight \nu on \mathbb Z^2 such that 1\leq \nu \leq \omega, \nu is nonconstant if and only if \omega is nonconstant and \sum _{\mathbf {m}\in \mathbb Z^2}\left |\widehat {(\frac {1}{f})}(\mathbf {m})\right |\nu (\mathbf {m})<\infty. If \varphi is holomorphic on a neighbourhood of the range of f, then there is weight \chi on \mathbb Z^2 such that 1\leq \chi \leq \omega, \chi is nonconstant if and only if \omega is nonconstant and \sum _{\mathbf {m}\in \mathbb Z^2}|\widehat {(\varphi \circ f)}(\mathbf {m})|\chi (\mathbf {m})<\infty. An analogue of Żelazko's theorem on p-th power convergence is also discussed. These two variables weighted analogues of classical theorems of Wiener, Lévy and Żelazko are proved exhibiting the complexity of weights in two variables. [ABSTRACT FROM AUTHOR]

Kida and Tucker-Drob recently extended the notion of inner amenability from countable groups to discrete p.m.p. groupoids. In this article, we show that inner amenable groupoids have "fixed priced 1" in the sense that every principal extension of an inner amenable groupoid has cost 1. This simultaneously generalizes and unifies two well known results on cost from the literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p. equivalence relation admitting a nontrivial asymptotically central sequence in its full group has cost 1, and (2) a theorem of Tucker-Drob stating that inner amenable groups have fixed price 1. [ABSTRACT FROM AUTHOR]

Let X be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base Q. A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space Y, which can be considered as a variant of the T-dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in X embeds fully faithfully in the derived category of (twisted) coherent sheaves on Y, under the technical assumption that π2(Q) vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras. [ABSTRACT FROM AUTHOR]

We prove a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in the sense of Nica, then so are its submonoids. As applications, we complete the classification of Artin monoids with respect to Nica amenability by showing that only the right-angled ones are amenable in the sense of Nica and we show that the Nica amenability of a graph product of right-LCM semigroups having no nontrivial units is inherited by the factors. [ABSTRACT FROM AUTHOR]

We prove that a symplectic matrix with entries in a ring with Bass stable rank one can be factored as a product of elementary symplectic matrices. This also holds for null-homotopic symplectic matrices with entries in a Banach algebra or in the ring of complex valued continuous functions on a finite dimensional normal topological space. [ABSTRACT FROM AUTHOR]

We give formulas for the spectral radius of weighted endomorphisms aα : C(X,D) → C(X,D), a ∈ C(X,D), where X is a compact Hausdorff space and D is a unital Banach algebra. Under the assumption that α generates a partial dynamical system (X,φ), we establish two kinds of variational principles for r(aα): using linear extensions of (X,φ) and using Lyapunov exponents associated with ergodic measures for (X,φ). This requires considering (twisted) cocycles over (X,φ) with values in an arbitrary Banach algebra D, and thus our analysis cannot be reduced to any of the multiplicative ergodic theorems known so far. The established variational principles apply not only to weighted endomorphisms but also to a vast class of operators acting on Banach spaces that we call abstract weighted shifts associated with α : C(X,D) → C(X,D). In particular, they are far-reaching generalizations of formulas obtained by Kitover, Lebedev, Latushkin, Stepin, and others. They are most efficient when D= B(F), for a Banach space F, and endomorphisms of B(F) induced by α are inner isometric. As a by-product we obtain a dynamical variational principle for an arbitrary operator b ∈ B(F) and that its spectral radius is always a Lyapunov exponent in some direction v ∈ F when F is reflexive. [ABSTRACT FROM AUTHOR]

We study the structure of one-sided ideals in a Banach algebra A. We find very general conditions under which any left (right) ideal is of the form Aq (qA) for some idempotent right (left) multiplier on A. We further show that a large class of one-sided multipliers can be realized as a product of an invertible and an idempotent multiplier. Applying our results to algebras over locally compact quantum groups and C*.-algebras, we demonstrate that our approach generalizes and unifies various theorems from abstract harmonic analysis and operator algebra theory. In particular, we generalize results of Bekka (and Reiter), Berglund, Forrest, and Lau. Losert. We also deduce the Choquet. Deny theorem for compact groups as an application of our approach. Moreover, we answer, for a certain class of measures on a compact group, a question of ¡§ Ulger which, in the abelian case, goes back to Beurling (1938). [ABSTRACT FROM AUTHOR]

For a nontrivial locally compact group G, consider the Banach algebras of p-pseudofunctions, p-pseudomeasures, p-convolvers, and the full group Lp-operator algebra. We show that these Banach algebras are operator algebras if and only if p = 2. More generally, we show that for q ∈ [1,∞), if any of these Banach algebras can be represented on an Lq-space, then one of the following holds: (a) p = 2 and G is abelian; or (b) |1/p - 1/2| = |1/q - 1/2|. This result can be interpreted as follows: for p, q ∈ [1,∞), the Lp- and Lq-representation theories of a group are incomparable, except in the trivial cases when they are equivalent. As an application, we show that, for distinct p, q ∈ [1,∞), if the Lp- and Lq-crossed products of a topological dynamical system are isomorphic, then 1/p + 1/q = 1. In order to prove this, we study the following relevant aspects of Lp-crossed products: existence of approximate identities, duality with respect to p, and existence of canonical isometric maps from group algebras into their multiplier algebras. [ABSTRACT FROM AUTHOR]

Denote by W1 the set of complex valued functions of the form a(z) = Σ+∞ i=-∞ aizi such that Σ+∞ i=-∞ ∣iai∣ < ∞. We call QT-matrix a quasi-Toeplitz matrix A, associated with a symbol a(z) ∈ W1, of the form A = T(a) + E, where T(a) = (ti,j )i,j∈Z+ is the semi-infinite Toeplitz matrix such that ti,j = aj-i, for i, j ∈ Z+, and E = (ei,j )i,j∈Z+ is a semi-infinite matrix such thatΣ+∞ i,j=1 ∣ei,j ∣ is finite. We prove that the class of QT-matrices is a Banach algebra with a suitable sub-multiplicative matrix norm. We introduce a finite representation of QT-matrices together with algorithms which implement elementary matrix operations. An application to solving quadratic matrix equations of the kind AX2 +BX +C = 0, encountered in the solution of Quasi-Birth and Death (QBD) stochastic processes with a denumerable set of phases, is presented where A,B,C are QT-matrices. [ABSTRACT FROM AUTHOR]

The purpose of this note is to show that if A is a Banach algebra with the continuous dual space A* and D : A → A* is a weakly compact derivation, then D** : A** → A*** is also a derivation, where A** has the first (or second) Arens product and A*** is viewed as the dual module of the Banach algebra A**. [ABSTRACT FROM AUTHOR]

It is shown that there exists a compact set X in CN (N ≥ 2) such that X̂\ X is nonempty and the uniform algebra P(X) has a dense set of invertible elements, a large Gleason part, and an abundance of nonzero bounded point derivations. The existence of a Swiss cheese X such that R(X) has a Gleason part of full planar measure and a nonzero bounded point derivation at almost every point is established. An analogous result in CN is presented. The analogue for rational hulls of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is established. The results presented address questions raised by Dales and Feinstein. [ABSTRACT FROM AUTHOR]

We give a simple and explicit description of the Bernstein-Szegö type measures associated with Jacobi matrices which differ from the Jacobi matrix of the Chebyshev measure in finitely many entries. We also introduce a class of measures M which parametrizes the Jacobi matrices with exponential decay and for each element in M we define a scattering function. Using Banach algebras associated with increasing Beurling weights, we prove that the exponential decay of the coefficients in a Jacobi matrix is completely determined by the decay of the negative Fourier coefficients of the scattering function. Combining this result with the Bernstein-Szegö type measures we provide different characterizations of the rate of decay of the entries of the Jacobi matrices for measures in M. [ABSTRACT FROM AUTHOR]

We prove that if an algebra is either selfinjective, local or graded, then the Hochschild homology dimension of its trivial extension is infinite. [ABSTRACT FROM AUTHOR]

Let λ be an infinite cardinal number and let ℓc ∞(λ) denote the subspace of ℓ∞(λ) consisting of all functions that assume at most countably many non-zero values. We classify all infinite-dimensional complemented subspaces of ℓc ∞(λ), proving that they are isomorphic to ℓc ∞(κ) for some cardinal number κ. Then we show that the Banach algebra of all bounded linear operators on ℓc ∞(λ) or ℓ∞(λ) has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws' approach description of the lattice of all closed ideals of B(X), where X = c0(λ) or X = ℓp(λ) for some p ∈ [1,∞), and we classify the closed ideals of B(ℓc ∞(λ)) that contains the ideal of weakly compact operators. [ABSTRACT FROM AUTHOR]

Using representation theory techniques we prove that various spaces of derivations or one-sided multipliers over certain operator algebras are reflexive. A sample result: any bounded local derivation (local left multiplier) on an automorphic semicrossed product C(Ω) ×σ Z+ is a derivation (resp. left multiplier). In the process we obtain various results of independent interest. In particular, we show that the finite dimensional nest representations of the tensor algebra of a topological graph separate points. [ABSTRACT FROM AUTHOR]

We establish subdifferential calculus rules for the sum of convex functions defined on normed spaces. This is achieved by means of a condition relying on the continuity behaviour of the inf-convolution of their corresponding conjugates, with respect to any given topology intermediate between the norm and the weak* topologies on the dual space. Such a condition turns out to also be necessary in Banach spaces. These results extend both the classical formulas by Hiriart-Urruty and Phelps and by Thibault. [ABSTRACT FROM AUTHOR]

We study semigroup C*-algebras of ax+b-semigroups over integral domains. The goal is to generalize several results about C*-algebras of ax+bsemigroups over rings of algebraic integers. We prove results concerning Ktheory and structural properties like the ideal structure or pure infiniteness. Our methods allow us to treat ax + b-semigroups over a large class of integral domains containing all noetherian, integrally closed domains and coordinate rings of affine varieties over infinite fields. [ABSTRACT FROM AUTHOR]

We show that if E is an atomic Banach lattice with an order continuous norm, A,B ∈ Lr(E) and MA,B is the operator on Lr (E) defined by MA,B(T) = ATB, then ||MA,B||r = ||A||r||B||r but that there is no real α > 0 such that ||MA,B|| ≥ α||A||r||B||r. [ABSTRACT FROM AUTHOR]

For compact metric spaces (X, d), we show that the Lipschitz spaces Lip(X, d) and the little Lipschitz spaces lip(X, dα) with 0 < α < 1, equipped with the sum norm, support only trivial hermitian operators, that is, real multiples of the identity operator. [ABSTRACT FROM AUTHOR]

We give a counterexample to the following theorem of Bremermann on Shilov boundaries: if D is a bounded domain in Ćn having a univalent envelope of holomorphy, say D, then the Shilov boundary of D with respect to the algebra A(D), call it ∂sD, coincides with the corresponding one for D, called ∂sD. [ABSTRACT FROM AUTHOR]