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Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 42 (12 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
H ∗ALGEBRAS AND QUANTIZATION OF PARAHERMITIAN SPACES
, 2006
"... Abstract. In the present note we describe a family of H ∗algebra structures on the set L 2 (X) of square integrable functions on a rankone paraHermitian symmetric space X. ..."
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Abstract. In the present note we describe a family of H ∗algebra structures on the set L 2 (X) of square integrable functions on a rankone paraHermitian symmetric space X.
AN ALGEBRAIC APPROACH TO WIGNER’S UNITARYANTIUNITARY THEOREM
, 1998
"... We present an operator algebraic approach to Wigner’s unitaryantiunitary theorem using some classical results from ring theory. To show how effective this approach is, we prove a generalization of this celebrated theorem for Hilbert modules over matrix algebras. We also present a Wignertype result ..."
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We present an operator algebraic approach to Wigner’s unitaryantiunitary theorem using some classical results from ring theory. To show how effective this approach is, we prove a generalization of this celebrated theorem for Hilbert modules over matrix algebras. We also present a Wignertype result for maps on prime C ∗algebras.
H*algebras and . . . first steps in infinitedimensional categorical quantum mechanics
, 2010
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AN EQUATION ON OPERATOR ALGEBRAS AND
"... Abstract. In this paper we prove the following result: Let X be a Banach space over the real or complex field F and let L(X) be the algebra of all bounded linear operators on X. Suppose there exists an additive mapping T: A(X) → L(X), where A(X) ⊂ L(X) is a standard operator algebra. Suppose that ..."
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Abstract. In this paper we prove the following result: Let X be a Banach space over the real or complex field F and let L(X) be the algebra of all bounded linear operators on X. Suppose there exists an additive mapping T: A(X) → L(X), where A(X) ⊂ L(X) is a standard operator algebra. Suppose that T (A3) = AT (A)A holds for all A ∈ A(X). In this case T is of the form T (A) = λA for any A ∈ A(X) and some λ ∈ F. This result is applied to semisimple H∗−algebras. This research is related to the work of Molnár [8] and is a continuation of our work [9, 10]. Throughout, R will represent an associative ring with center Z(R). A ring R is ntorsion free, where n> 1 is an integer, if nx = 0, x ∈ R implies x = 0. The commutator xy − yx will be denoted by [x, y]. We shall use basic commutator identities [xy, z] = [x, z]y + x[y, z] and [x, yz] = [x, y] z + y [x, z]. Recall that R is prime if aRb = (0) implies a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping D: R → R is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R and is called a Jordan derivation in case D(x 2) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R, such that D(x) = [a, x] holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [6] asserts that any Jordan derivation on a 2torsion free prime ring is a derivation. A brief proof of Herstein’s result can be found in [3]. Cusack [5] generalized Herstein’s result to 2−torsion free semiprime rings (see also [2] for an alternative proof).