Quasi-Radial Quasi-Homogeneous Symbols and Commutative Banach Algebras of Toeplitz Operators

We present here a quite unexpected result: Apart from already known commutative C*-algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C*-algebra, and for n = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols.     

Spectral invariance of smooth crossed products, and rapid decay locally compact groups

We show that all rapid-decay locally compact groups are unimodular and that the set of rapid-decay functions on a locally compact rapidly decaying group forms a dense and spectral invariant Fréchet *-subalgebra of the reduced group C ^*-algebra. In general, the set of rapid-decay functions on a locally compact strongly rapid-decay group with values in a commutative C ^*-algebra forms a dense and spectral invariant Fréchet *-subalgebra of the twisted crossed product C ^*-algebra. The spectral invariance property implies that the K-theories of both algebras are naturally isomorphic under inclusion.This project is supported in part by the National Science Foundation Grant #DMS 92-04005.     

Commutative <Emphasis Type="Italic">C</Emphasis>*-Algebras of Toeplitz Operators on Complex Projective Spaces

We prove the existence of commutative C*-algebras of Toeplitz operators on every weighted Bergman space over the complex projective space \mathbbPⁿ\mathbb(C){{\mathbb{P}^n}\mathbb{(C)}}. The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an associated pair of Lagrangian foliations with distinguished geometric properties and are closely related to the geometry of \mathbbPⁿ\mathbb(C){{\mathbb{P}^n}\mathbb{(C)}}.     

Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action

Studying commutative C*-algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C*-algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each standard weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent and quasi-nilpotent. Recently it was observed in Vasilevski (Integr Equ Oper Theory. 66:141–152, 2010) that there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach, and being extended to C*-algebras they became non-commutative. These results were extended then to the classes of symbols, subordinated to the quasi-hyperbolic and quasi-parabolic groups. In this paper we prove the analogous commutativity result for Toeplitz operators whose symbols are subordinated to the quasi-nilpotent group. At the same time we conjecture that apart from the known C*-algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group and which are commutative on each weighted Bergman space.     

Unitary stable ranks and norm-one ranks

In the context of commutative C*-algebras, we solve a problem related to a question of M. Rieffel by showing that the all-units rank and the norm-one rank coincide with the topological stable rank. We also introduce the notion of unitary M-stable rank for an arbitrary commutative unital ring and compare it with the Bass stable rank. In case of uniform algebras, a su?cient condition for norm-one reducibility is given.     

Reflexivity of the automorphism and isometry groups of C*-algebras in BDF theory

We prove that the automorphism and isometry groups of any extension of the C*-algebra C (H)\cal C (\cal H) of all compact operators by a separable commutative C*-algebra are algebraically reflexive. Concerning the possibly most important extensions by the algebra C(\Bbb T)C(\Bbb T) of all continuous complex valued functions on the perimeter of the unit disc, we show that these groups are topologically nonreflexive.     

Bounded and Unitary Elements in Pro-C*-algebras

A pro-C*-algebra is a (projective) limit of C*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C*-algebras can be seen as non-commutative k-spaces. An element of a pro-C*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous *-homomorphism into a C*-algebra. The *-subalgebra consisting of the bounded elements turns out to be a C*-algebra. In this paper, we investigate pro-C*-algebras from a categorical point of view. We study the functor (−) _b that assigns to a pro-C*-algebra the C*-algebra of its bounded elements, which is the dual of the Stone-Čech-compactification. We show that (−) _b is a coreflector, and it preserves exact sequences. A generalization of the Gelfand duality for commutative unital pro-C*-algebras is also presented.     

The category of commutative HopfC *-algebras

In this paper, we first continue our study of group duality, and prove that the duality we established earlier is natural. Then we use this naturality to study the category of commutative, cocommutative HopfC ^*-algebras, and show that the category of compact Abelian semigroups and the category of commutative, cocommutative HopfC ^*-algebras with units are isomorphic. By using this result, we show that the category of commutative, cocommutative quantum groups is Abelian. This is a generalization of a result of Grothendieck about the catrgory of finite-dimensional commutative, cocommutative Hopf algebras with antipodes.     

The Bunce-Deddens algebras as crossed products by partial automorphisms

We describe both the Bunce-DeddensC ^*-algebras and their Toeplitz versions, as crossed products of commutativeC ^*-algebras by partial automorphisms. In the latter case, the commutative algebra has, as its spectrum, the union of the Cantor set and a copy of the set of natural numbers , fitted together in such a way that is an open dense subset. The partial automorphism is induced by a map that acts like the odometer map on the Cantor set while being the translation by one on . From this we deduce, by taking quotients, that the Bunce-DeddensC ^*-algebras are isomorphic to the (classical) crossed product of the algebra of continuous functions on the Cantor set by the odometer map.     

Polynomially dependent homomorphisms and frobenius n-homomorphisms

We introduce and study polynomially dependent homomorphisms, which are special linear maps between associative algebras with identity. The multiplicative structure is much involved in the definition of such homomorphisms (we consider only the case of maps f: A → B with commutative B). The most important particular case of these maps are the Frobenius n-homomorphisms, which were introduced by V.M. Buchstaber and E.G. Rees in 1996–1997. A 1-homomorphism f: A → B is just an algebra homomorphism (the algebra B is commutative). A typical example of an n-homomorphism is given by the sum of n algebra homomorphisms, f = f ₁ + ... + f _n , f _i : A → B, 1 ≤ i ≤ n. Another example is the trace of n × n matrices over a field R of characteristic zero, tr: M _n (R) → R, and, more generally, the character of any n-dimensional representation, tr ρ: A → R, ρ: A → M _n (R). The properties of n-homomorphisms (some of which were proved by Buchstaber and Rees under additional conditions) are derived, and a general theory of polynomially dependent homomorphisms is developed. One of the main results of the paper is a uniqueness theorem, which distinguishes the classes of n-homomorphisms among all polynomially dependent homomorphisms by a single natural completeness condition. As a topological application of n-homomorphisms, we consider the theory of n-homomorphisms between commutative C*-algebras with identity. We prove that the norm of any such n-homomorphism is equal to n and describe the structure of all such n-homomorphisms, which generalizes the classical Gelfand transform (the case of n = 1). An interesting fact discovered is that the Gelfand transform, which is a functorial bijection between appropriate spaces of maps, becomes a homeomorphism after considering natural topologies on these spaces.     

Spherical Harmonic Analysis on Buildings of Type ? n

 To any locally finite thick building of type there is naturally associated a commutative algebra of operators. When is constructed from a local field F with local ring , and , then is isomorphic to the convolution algebra of compactly supported bi-K-invariant functions on PGL(n+1,F). We give a proof, valid for any , that the multiplicative functionals on may all be expressed in terms of Hall–Littlewood polynomials. Regarding as a subalgebra of the C ^*-algebra of bounded operators on the space of square summable functions on the vertex set of , we find the spectrum of the C ^*-algebra , the closure of . This generalizes results obtained in [3] when n = 1 and in [5] when n = 2.     

Quantum stochastic convolution cocycles III

Every Markov-regular quantum Lévy process on a multiplier C ^*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C ^*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Lévy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C ^*-bialgebra, to locally compact quantum groups and multiplier C ^*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.     

Primary Decompositions in Varieties of Commutative Diassociative Loops

The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all nth powers are central, for a fixed n. For n = 2, we get precisely commutative C loops. For n = 3, a prominent variety is that of commutative Moufang loops.

Many analogies between commutative C and Moufang loops have been noted in the literature, often obtained by interchanging the role of the primes 2 and 3. We show that the correct encompassing variety for these two classes of loops is the variety of commutative RIF loops. In particular, when Q is a commutative RIF loop: all squares in Q are Moufang elements, all cubes are C elements, Moufang elements of Q form a normal subloop M ₀(Q) such that Q/M ₀(Q) is a C loop of exponent 2 (a Steiner loop), C elements of L form a normal subloop C ₀(Q) such that Q/C ₀(Q) is a Moufang loop of exponent 3. Since squares (resp., cubes) are central in commutative C (resp., Moufang) loops, it follows that Q modulo its center is of exponent 6. Returning to the decomposition theorem, we find that every torsion, commutative RIF loop is a direct product of a C 2-loop, a Moufang 3-loop, and an abelian group with each element of order prime to 6.

We also discuss the definition of Moufang elements and the quasigroups associated with commutative RIF loops.     

Spherical Harmonic Analysis on Buildings of Type à n

 To any locally finite thick building of type there is naturally associated a commutative algebra of operators. When is constructed from a local field F with local ring , and , then is isomorphic to the convolution algebra of compactly supported bi-K-invariant functions on PGL(n+1,F). We give a proof, valid for any , that the multiplicative functionals on may all be expressed in terms of Hall–Littlewood polynomials. Regarding as a subalgebra of the C ^*-algebra of bounded operators on the space of square summable functions on the vertex set of , we find the spectrum of the C ^*-algebra , the closure of . This generalizes results obtained in [3] when n = 1 and in [5] when n = 2. (Received 26 June 2000; in revised form 21 February 2001)     

A set of maps from K to EndA(l2(A)) isomorphic to EndA(K)(l2(A(K))). Applikations

Let A be a C^*-algebra, K be a compact space, A(K) be the C^*-algebra of all continuous maps from K into A, 1₂(A) be the standard countably generated Hilbert A-module. We investigate a set of maps from K into End_A(1₂(A)), which is isomorphic to End_A(K)(1₂(A(K))). We describe the subsets which are isomorphic to End_fA(K) ^*(1₂(A(K))). GL_A(K)(1₂(A(K))) and GL_fA(K) ^*(1₂(A(K))), respectively. As an application we deduce a criterion for the self-duality of 1₂(A) in the commutative case.     

The Manifold-Valued Dirichlet Problem for Symmetric Diffusions

Harmonic maps between two Riemannian manifolds M and N are often constructed as energy minimizing maps. This construction is extended for the Dirichlet problem to the case where the Riemannian energy functional on M is replaced by a more general Dirichlet form. We obtain weakly harmonic maps and prove that these maps send the diffusion to N-valued martingales. The basic tools are the reflected Dirichlet space and the stochastic calculus for Dirichlet processes.     

A complementary universal conjugate Banach space and its relation to the approximation problem

LetC ₁=(ΣG _n ) _{l 1}, where (G _n ) is a sequence which is dense (in the Banach-Mazur sense) in the class of all finite dimensional Banach spaces. IfX is a separable Banach space, thenX ^* is isometric to a subspace ofC ₁ ^* =(ΣG _n ^* ) _m which is the range of a contractive projection onC ₁ ^* . Separable Banach spaces whose conjugates are isomorphic toC ₁ ^* are classified as those spaces which contain complemented copies of C₁. Applications are that every Banach space has the [metric] approximation property ([m.] a.p., in short) iff (ΣG _n ^* ) _m does, and if there is a space failing the m.a.p., thenC ₁ can be equivalently normed to fail the m.a.p. The author was supported in part by NSF GP 28719.     

On C *-Extreme Maps and *-Homomorphisms of a Commutative C *-Algebra

The generalized state space of a commutative C^*-algebra, denoted , is the set of positive unital maps from C(X) to the algebra of bounded linear operators on a Hilbert space . C^*-convexity is one of several non-commutative analogs of convexity which have been discussed in this context. In this paper we show that a C^*-extreme point of satisfies a certain spectral condition on the operators in the range of the associated positive operator-valued measure. This result enables us to show that C^*-extreme maps from C(X) into , the algebra generated by the compact and scalar operators, are multiplicative. This generalizes a result of D. Farenick and P. Morenz. We then determine the structure of these maps. This paper constitutes a part of the author’s Ph.D. thesis at the University of Nebraska-Lincoln.     

The Algebraic K-Theory of Operator Algebras

We the study the algebraic K-theory of C *-algebras, forgetting the topology. The main results include a proof that commutative C*-algebras are K-regular in all degrees (that is, all theirN ^T K _iand extensions of the Fischer-Prasolov Theorem comparing algebraic and topological K-theory with finite coefficients.