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Involutive categories and monoids, with a GNScorrespondence
 In Quantum Physics and Logic (QPL
, 2010
"... This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of EilenbergMoore algebras of involutive monads are involutive, with conjugation for modules and vecto ..."
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This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of EilenbergMoore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the socalled GelfandNaimarkSegal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive symmetric monoidal categories. 1
Notes on normed algebras
, 2004
"... All vector spaces and so forth here will be defined over the complex numbers. If z = x+i y is a complex number, where x, y are real numbers, then the complex conjugate of z is denoted z and defined to be x − i y. The complex conjugate of a sum or product of complex numbers is equal to the correspond ..."
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All vector spaces and so forth here will be defined over the complex numbers. If z = x+i y is a complex number, where x, y are real numbers, then the complex conjugate of z is denoted z and defined to be x − i y. The complex conjugate of a sum or product of complex numbers is equal to the corresponding sum or product of complex conjugates. The modulus of a complex number z is the nonnegative real number z  such that z  2 is equal to the product of z and its complex conjugate. Thus the modulus of a product of complex numbers is equal to the product of their moduli, and one can show that the modulus of a sum of two complex numbers is less than or equal to the sum of the moduli of the complex numbers. By a finitedimensional algebra we mean a finite dimensional complex vector space A equipped with a binary operation which satisfies the usual associativity and distributivity properties, and which has a nonzero multiplicative identity element e. In other words, e x = xe = x for all x ∈ A. Thus A should have positive dimension in particular. Notice that the multiplicative
Notes on normed algebras, 4
, 2004
"... Let A be a finitedimensional algebra over the complex numbers with nonzero identity element e. If x ∈ A, then the resolvent set associated to x is the set ρ(x) of complex numbers λ such that λ e − x is invertible, and the spectrum of x is the set σ(x) of complex numbers λ such that λ e − x is not i ..."
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Let A be a finitedimensional algebra over the complex numbers with nonzero identity element e. If x ∈ A, then the resolvent set associated to x is the set ρ(x) of complex numbers λ such that λ e − x is invertible, and the spectrum of x is the set σ(x) of complex numbers λ such that λ e − x is not invertible. For instance, if V is a finitedimensional vector space over the complex numbers of positive dimension, L(V) is the algebra of linear operators on V, and T is a linear operator on V, then a complex number λ lies in the spectrum of T if and only if λ I − T has a nontrivial kernel. This is equivalent to saying that there is a nonzero vector v ∈ V such that T(v) = λ v, which is to say that v is a nonzero eigenvector for T with eigenvalue λ. Let p(z) be a polynomial on the complex numbers, which can be written explicitly as (1) p(z) = cm z m + cm−1 z m−1 + · · · + c0 for some complex numbers c0,...,cm. If A is a finitedimensional algebra
Quasifractal sets in space
, 709
"... Let a be a positive real number, a < 1/2. A standard construction of a selfsimilar Cantor set in the plane starts with the unit square [0,1] × [0,1], replaces it with the four corner squares with sidelength a, then replaces each of those squares with their four corner squares of sidelength a 2, an ..."
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Let a be a positive real number, a < 1/2. A standard construction of a selfsimilar Cantor set in the plane starts with the unit square [0,1] × [0,1], replaces it with the four corner squares with sidelength a, then replaces each of those squares with their four corner squares of sidelength a 2, and so on. At the nth stage one has 4 n squares with sidelength a −n, and the resulting Cantor set has Hausdorff dimension log 4/( − log a). The limiting case a = 1/2 simply reproduces the unit square, which has Hausdorff dimension 2. Suppose that we keep the boundaries of the squares at each stage of the construction, to get a kind of quasifractal set consisting of the Cantor set and a countable collection of line segments. The sum of the lengths of these line segments is finite exactly when a < 1/4. The Cantor set may be described as the singular part of this quasifractal set, which is compact and connected. Of course, one can consider similar constructions in higher dimensions. For the sake of simplicity, let us focus on connected fractal sets in R 3 with
Analysis on disconnected sets
, 709
"... Very often in analysis, one focuses on connected spaces. This is certainly not always the case, and in particular there are many interesting matters related to Cantor sets. Here we are more concerned with a type of complementary situation. As a basic scenario, suppose that U is an open set in R n an ..."
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Very often in analysis, one focuses on connected spaces. This is certainly not always the case, and in particular there are many interesting matters related to Cantor sets. Here we are more concerned with a type of complementary situation. As a basic scenario, suppose that U is an open set in R n and that E is a closed set contained in the boundary of U such that for every x∈U and r> 0 there is a connected component of U contained in the ball with center x and radius r. For instance, E might be the boundary of U. As a uniform version of the condition, one might ask that there be a constant C> 0 such that the aforementioned connected component of U contains a ball of radius C −1 r. The connected components of U might be quite regular, even if there are infinitely many of them. As a basic example, E could be a Cantor set in the real line, and U could be the complement of E or the union of the bounded complementary components of E. This does not work in higher dimensions, where the complement of a Cantor set is connected. Sierpinski gaskets and carpets in the plane are
MODEL THEORY OF A NONDEGENERATE
"... Abstract. We study the theory of a Hilbert space H as a module for a unital C∗algebra A from the point of view of continuous logic. We show this theory, in an appropiate lenguage, has quantifier elimination and it is superstable. We show that for every v ∈ H the type tp(v/∅) is in correspondence wi ..."
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Abstract. We study the theory of a Hilbert space H as a module for a unital C∗algebra A from the point of view of continuous logic. We show this theory, in an appropiate lenguage, has quantifier elimination and it is superstable. We show that for every v ∈ H the type tp(v/∅) is in correspondence with the positive linear functional over A defined by v. Finally, we characterize forking, orthogonality and domination of types and show the theory has weak elimination of imaginaries. 1. introduction Let A be a unital C ∗algebra and let π: A → B(H) be a C ∗algebra nondegenerate isometric homomorphism, where B(H) is the algebra of bounded operators over a Hilbert space H. The goal of this paper is to study H as a metric structure expanded by A from the point of view of continuous logic (see [6] and [5]). In order to describe the structure of H as a module for A, we include a symbol ˙a in the language of the Hilbert space structure whose interpretation in H will be π(a) for every a in the unit ball of A. Following [5], we study the theory of H as a metric
(March 4, 2012) Compact operators on Banach spaces: FredholmRiesz
"... [Fredholm 1900/1903] treated compact operators as limiting cases of finiterank operators. [1] [Riesz 1917] defined and made direct use of the compactness condition, more apt for Banach spaces. See [RieszNagy 1952] for extensive discussion in the Hilbertspace situation, and many references to orig ..."
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[Fredholm 1900/1903] treated compact operators as limiting cases of finiterank operators. [1] [Riesz 1917] defined and made direct use of the compactness condition, more apt for Banach spaces. See [RieszNagy 1952] for extensive discussion in the Hilbertspace situation, and many references to original papers. [Arveson 2002] gives a succinct discussion. Here, we prove the basic Fredholm alternative on Banach spaces, that for compact T and nonzero λ ∈ C, either T − λ is a bijection, or has closed image of codimension equal to the dimension of its kernel. In particular, the only nonzero spectrum is point spectrum. [2] A special case of this is widely useful in perturbation theory: it is often important to know, for λ = 0 not an eigenvalue of compact T: X → X, that T − λ is surjective. In the appendix, we recall the equivalence of precompactness and total boundedness in complete metric spaces, and recall the ArzelaAscoli result, that the precompact subset of C o (X) for compact X consist of equicontinuous, uniformly bounded collections of functions. 1. Compact operators on Banach spaces [1.1] Notion of compact operator A continuous linear operator T: X → Y on Banach spaces is compact when T maps bounded sets in X to