The Hilbert space L2(SU(2)) as a representation space for the group (SU(2) × SU(2)) Ⓢ S2

A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972719000042 ◽

2019 ◽

Vol 100 (2) ◽

pp. 317-322

Author(s):

H. KUMUDINI DHARMADASA ◽

WILLIAM MORAN

Keyword(s):

Hilbert Space ◽

Invariant Measure ◽

Banach Spaces ◽

Closed Subgroup ◽

General Setting ◽

Locally Compact Group ◽

Reciprocity Theorem ◽

Representation Space ◽

Locally Compact ◽

Frobenius Reciprocity

Let $G$ be a locally compact group and $K$ a closed subgroup of $G$. Let $\unicode[STIX]{x1D6FE},$$\unicode[STIX]{x1D70B}$ be representations of $K$ and $G$ respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space $G/K$ possesses a right-invariant measure and the representation space $H(\unicode[STIX]{x1D6FE})$ of the representation $\unicode[STIX]{x1D6FE}$ of $K$ is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of a quasi-invariant measure on $G/K$ and that the representation spaces $\mathfrak{B}(\unicode[STIX]{x1D6FE})$ and $\mathfrak{B}(\unicode[STIX]{x1D70B})$ are Banach spaces with $\mathfrak{B}(\unicode[STIX]{x1D70B})$ being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.

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An Extension of the Concept of the Order Dual of a Riesz Space

Canadian Journal of Mathematics ◽

10.4153/cjm-1967-041-6 ◽

1967 ◽

Vol 19 ◽

pp. 488-498 ◽

Cited By ~ 9

Author(s):

W. A. J. Luxemburg ◽

J. J. Masterson

Keyword(s):

Hilbert Space ◽

Topological Space ◽

Hausdorff Space ◽

Dense Subset ◽

Multiplication Operator ◽

Closed Operator ◽

Riesz Space ◽

Open Dense Subset ◽

Representation Space ◽

Totally Disconnected

Let L be a σ-Dedekind complete Riesz space. In (8), H. Nakano uses an extension of the multiplication operator on a Riesz space into itself (analagous to the closed operator on a Hilbert space) to obtain a representation space for the Riesz space L. He calls such an operator a “dilatator operator on L.” More specifically, he shows that the set of all dilatator operators , when suitable operations are defined, is a Dedekind complete Riesz space which is isomorphic to the space of all functions defined and continuous on an open dense subset of some fixed totally disconnected Hausdorff space. The embedding of L in the function space is then obtained by showing that L is isomorphic to a Riesz subspace of . Moreover, when L is Dedekind complete, it is an ideal in , and the topological space is extremally disconnected.

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C2 TODA THEORY IN THE REDUCED WZNW FRAMEWORK

International Journal of Modern Physics A ◽

10.1142/s0217751x94002168 ◽

1994 ◽

Vol 09 (30) ◽

pp. 5387-5407 ◽

Cited By ~ 3

Author(s):

Z. BAJNOK

Keyword(s):

Hilbert Space ◽

Symmetry Algebra ◽

Equation Of Motion ◽

Representation Space ◽

Classical Representation ◽

Selection Rules ◽

Highest Weight ◽

Highest Weight Representations

We consider the C2 Toda theory in the reduced WZNW framework. Analyzing the classical representation space of the symmetry algebra (which is the corresponding C2W algebra), we determine its classical highest weight representations. We quantize the model, promoting only the relevant quantities to operators. Using the quantized equation of motion we determine the selection rules for the u field, which corresponds to one of the Toda fields, and we give restrictions for its amplitude functions and for the structure of the Hilbert space of the model.

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