General Theory of Banach Algebra. By C.E. Rickart. Pp. 394. 79s. 1960. (Van Nostrand, London)

Preliminaries.- Modular Hilbert algebras.- Generalized Hilbert algebras.- The commutation theorem for modular Hilbert algebras.- Self-adjoint subalgebras of generalized Hilbert algebras.- The spectral algebra.- The modular operator ?.- The resolvent of the modular operator ?.- The one-parameter automorphisms defined by the modular operator ?.- Formulation of the modular Hilbert algebra.- Tensor product and direct sum of modular Hilbert algebras.- The standard representation of von Neumann algebras.- The modular automorphism group and the Kubo-Martin-Schwinger boundary condition.- Semi-finiteness and the modular automorphism group.

Tomita's theory of modular Hilbert algebras and its applications

Inverse limits of C^*-algebras

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Vector bundles and K-theory over topological algebras

Introduction. Part I: General Theory. I. Background material. II. Locally C* -algebras. III. Representation theory. IV. Structure space of an m* -convex algebra. V. Hermitian and symmetric topological *-algebras. Part II: Applications. VI. Integral representations. Uniqueness of topology. VII. Tensor products of topological *-algebras. Bibliography.

Topological Algebras with Involution

/pdf/spaces-of-representations-and-enveloping-l-m-c-ast-algebras-39e4ekaxjb.pdf

Spaces of representations and enveloping l.m.c. $^{\ast}$-algebras.

V. Ptak's inequality is valid for every hermitian completeQ locallym-convex (:l.m.c.) algebra. Every algebra of the last kind is, in particular, symmetric. Besides, a (Hausdorff) locallyC*-algebra (being always symmetric) with the propertyQ is, within a topological algebraic isomorphism, aC*-algebra. Furthermore, a type of Raikov's criterion for symmetry is also valid for non-normed topological*-algebras. Concerning topological tensor products, one gets that symmetry of theπ-completed tensor product of two unital Frechet l.m.c.*-algebrasE, F (π denotes the projective tensorial topology) is always passed toE, F, while the converse occurs when moreover either ofE, F is commutative.

Symmetric topological*-algebras

Locally convex quasi *-algebras with sufficiently many *-representations

There are examples of C ⁄ -algebras A that accept a locally con- vex ⁄-topology t coarser than the given one, such that e A(t) (the completion of A with respect to t) to be a GB ⁄ -algebra. The multiplication of A(t) may be or not be jointly continuous. In the second case, e A(t) may fail being a lo- cally convex ⁄-algebra, but it is a partial ⁄-algebra. In both cases the structure and the representation theory of e A(t) are investigated. If A+ t denotes the t-closure of the positive cone A+ of the given C ⁄ -algebra A, then the prop- erty A+ t T (iA+ t ) = f0g is decisive for the existence of certain faithful ⁄- representations of the corresponding ⁄-algebra e A(t).

http://www1.unipa.it/fabio.bagarello/ricerca/articolipdf/completionBFIT_JOT.pdf

Maria Fragoulopoulou

Papers

Topological Algebras with Involution

Spaces of representations and enveloping l.m.c. $^{\ast}$-algebras.

Symmetric topological*-algebras

Locally convex quasi -algebras with sufficiently many -representations

The completion of a c ⁄ -algebra with a locally convex topology