Results 1 
4 of
4
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 ..."
Abstract

Cited by 43 (13 self)
 Add to MetaCart
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. ..."
Abstract
 Add to MetaCart
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 ..."
Abstract
 Add to MetaCart
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.