Toeplitz C*-Algebras on Super-Cartan Domains

Abstract

We study Hilbert spaces of super-holomorphic functions (including anti-commuting Grassmann variables) in the setting of bounded symmetric domains, more precisely for the matrix ball of arbitrary size. Our main results concern the classification of irreducible representations of the associated Toeplitz C * -algebra and an explicit decomposition of the super-Bergman space as a direct sum of vector-valued (ordinary) Bergman spaces.

Supersymmetric associated vector coherent states and generalized Landau levels arising from two-dimensional supersymmetry

“…Our construction also makes contact with another suggestion that has recently been made in the literature, which explicitly uses anticommuting Grassmann variables, 8,16 to introduce a quantization using super Toeplitz operators. The first is the construction of vector coherent states ͑VCSs͒, associated with supersymmetric ͑SUSY͒ pairs of Hamiltonians and the second is a generalization of the concept of Landau levels in the fractional quantum Hall effect ͑QHE͒, via SUSY pairs of Hamiltonians.…”

Commutative C*-Algebras Generated by Toeplitz Operators on the Super Unit Ball

“…In [5] the authors proved that every super Toeplitz operator T ν F on H 2 ν (B p|q ) is given by the 2 q × 2 q -matrix. With respect to the decomposition…”

“…In [5] the authors proved that the super Bergman space has a decomposition in direct sum of classical Bergman space and gave an explicit expression for the super Bergman projection. If we take Ψ = M⊂Q ψ M ξ M ∈ H 2 ν (B p|q ) ⊂ O(B p|q ) then the inner product has the form Γ(ν)…”

“…In [5] the authors proved that every super Toeplitz operator T ν F on H 2 ν (B p|q ) is given by the 2 q × 2 q -matrix. With respect to the decomposition Here, for 0 ≤ i, j ≤ q, T ν+j ν+i denotes the Bergman-type Toeplitz operator…”

“…These quantized supermanifolds yield the C * -algebra generated by such super Teoplitz operators. Along these lines, in [5] it is given an exhaustive description of the super Toeplitz operators over the super ball using classical Toeplitz-type operators.…”